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Theorem fprodabs 14543
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:   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍
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 2698 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 6557 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 14490 . . . . . 6 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
65fveq2d 6107 . . . . 5 (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
74prodeq1d 14490 . . . . 5 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
86, 7eqeq12d 2625 . . . 4 (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
98imbi2d 329 . . 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 6557 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 14490 . . . . . 6 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
1211fveq2d 6107 . . . . 5 (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
1310prodeq1d 14490 . . . . 5 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
1412, 13eqeq12d 2625 . . . 4 (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))
1514imbi2d 329 . . 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 6557 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 14490 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
1817fveq2d 6107 . . . . 5 (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
1916prodeq1d 14490 . . . . 5 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
2018, 19eqeq12d 2625 . . . 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 329 . . 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 6557 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 14490 . . . . . 6 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
2423fveq2d 6107 . . . . 5 (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
2522prodeq1d 14490 . . . . 5 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
2624, 25eqeq12d 2625 . . . 4 (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
2726imbi2d 329 . . 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 6142 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
2928adantl 481 . . . . 5 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
30 fzsn 12254 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3130adantl 481 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3231prodeq1d 14490 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33 simpr 476 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34 uzid 11578 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3534, 2syl6eleqr 2699 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀𝑍)
36 fprodabs.3 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
3736ralrimiva 2949 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
38 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑘𝑀 / 𝑘𝐴
3938nfel1 2765 . . . . . . . . . . . . 13 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
40 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
4140eleq1d 2672 . . . . . . . . . . . . 13 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
4239, 41rspc 3276 . . . . . . . . . . . 12 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
4337, 42mpan9 485 . . . . . . . . . . 11 ((𝜑𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4435, 43sylan2 490 . . . . . . . . . 10 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4544abscld 14023 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℝ)
4645recnd 9947 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℂ)
4729, 46eqeltrd 2688 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ)
48 prodsns 14541 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
4933, 47, 48syl2anc 691 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5032, 49eqtrd 2644 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5130prodeq1d 14490 . . . . . . . 8 (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
5251adantl 481 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53 prodsns 14541 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5433, 44, 53syl2anc 691 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5552, 54eqtrd 2644 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = 𝑀 / 𝑘𝐴)
5655fveq2d 6107 . . . . 5 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘𝑀 / 𝑘𝐴))
5729, 50, 563eqtr4rd 2655 . . . 4 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
5857expcom 450 . . 3 (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59 simp3 1056 . . . . . . . 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 6577 . . . . . . . . . . 11 (𝑛 + 1) ∈ V
61 csbfv2g 6142 . . . . . . . . . . 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 2619 . . . . . . . . 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 6567 . . . . . . 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 476 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
67 elfzuz 12209 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ𝑀))
6867, 2syl6eleqr 2699 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘𝑍)
6968, 36sylan2 490 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7069adantlr 747 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7166, 70fprodp1s 14540 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴))
7271fveq2d 6107 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)))
73 fzfid 12634 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑀...𝑛) ∈ Fin)
74 elfzuz 12209 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7574, 2syl6eleqr 2699 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7675, 36sylan2 490 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7776adantlr 747 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
7873, 77fprodcl 14521 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
79 peano2uz 11617 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
8079, 2syl6eleqr 2699 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
81 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑘(𝑛 + 1) / 𝑘𝐴
8281nfel1 2765 . . . . . . . . . . . . 13 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
83 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8483eleq1d 2672 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8582, 84rspc 3276 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8637, 85mpan9 485 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8780, 86sylan2 490 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
8878, 87absmuld 14041 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
8972, 88eqtrd 2644 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
90893adant3 1074 . . . . . . 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 14023 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
9291recnd 9947 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
9366, 92fprodp1s 14540 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · (𝑛 + 1) / 𝑘(abs‘𝐴)))
94933adant3 1074 . . . . . . 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 2654 . . . . . 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 1256 . . . . 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 11622 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
1003, 99mpcom 37 1 (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  csb 3499  {csn 4125  cfv 5804  (class class class)co 6549  cc 9813  1c1 9816   + caddc 9818   · cmul 9820  cz 11254  cuz 11563  ...cfz 12197  abscabs 13822  cprod 14474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-prod 14475
This theorem is referenced by:  etransclem23  39150
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