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Theorem fprodabs 11619
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 2270 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
4 oveq2 5882 . . . . . . 7 (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
54prodeq1d 11567 . . . . . 6 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑀)𝐴)
65fveq2d 5519 . . . . 5 (𝑎 = 𝑀 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴))
74prodeq1d 11567 . . . . 5 (𝑎 = 𝑀 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
86, 7eqeq12d 2192 . . . 4 (𝑎 = 𝑀 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
98imbi2d 230 . . 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 5882 . . . . . . 7 (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
1110prodeq1d 11567 . . . . . 6 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑛)𝐴)
1211fveq2d 5519 . . . . 5 (𝑎 = 𝑛 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴))
1310prodeq1d 11567 . . . . 5 (𝑎 = 𝑛 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
1412, 13eqeq12d 2192 . . . 4 (𝑎 = 𝑛 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)))
1514imbi2d 230 . . 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 5882 . . . . . . 7 (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
1716prodeq1d 11567 . . . . . 6 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴)
1817fveq2d 5519 . . . . 5 (𝑎 = (𝑛 + 1) → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴))
1916prodeq1d 11567 . . . . 5 (𝑎 = (𝑛 + 1) → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴))
2018, 19eqeq12d 2192 . . . 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 230 . . 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 5882 . . . . . . 7 (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
2322prodeq1d 11567 . . . . . 6 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)𝐴 = ∏𝑘 ∈ (𝑀...𝑁)𝐴)
2423fveq2d 5519 . . . . 5 (𝑎 = 𝑁 → (abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴))
2522prodeq1d 11567 . . . . 5 (𝑎 = 𝑁 → ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
2624, 25eqeq12d 2192 . . . 4 (𝑎 = 𝑁 → ((abs‘∏𝑘 ∈ (𝑀...𝑎)𝐴) = ∏𝑘 ∈ (𝑀...𝑎)(abs‘𝐴) ↔ (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
2726imbi2d 230 . . 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 5552 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
2928adantl 277 . . . . 5 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) = (abs‘𝑀 / 𝑘𝐴))
30 fzsn 10063 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3130adantl 277 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
3231prodeq1d 11567 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = ∏𝑘 ∈ {𝑀} (abs‘𝐴))
33 simpr 110 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
34 uzid 9540 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3534, 2eleqtrrdi 2271 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀𝑍)
36 fprodabs.3 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)
3736ralrimiva 2550 . . . . . . . . . . . 12 (𝜑 → ∀𝑘𝑍 𝐴 ∈ ℂ)
38 nfcsb1v 3090 . . . . . . . . . . . . . 14 𝑘𝑀 / 𝑘𝐴
3938nfel1 2330 . . . . . . . . . . . . 13 𝑘𝑀 / 𝑘𝐴 ∈ ℂ
40 csbeq1a 3066 . . . . . . . . . . . . . 14 (𝑘 = 𝑀𝐴 = 𝑀 / 𝑘𝐴)
4140eleq1d 2246 . . . . . . . . . . . . 13 (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝑀 / 𝑘𝐴 ∈ ℂ))
4239, 41rspc 2835 . . . . . . . . . . . 12 (𝑀𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → 𝑀 / 𝑘𝐴 ∈ ℂ))
4337, 42mpan9 281 . . . . . . . . . . 11 ((𝜑𝑀𝑍) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4435, 43sylan2 286 . . . . . . . . . 10 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘𝐴 ∈ ℂ)
4544abscld 11185 . . . . . . . . 9 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℝ)
4645recnd 7984 . . . . . . . 8 ((𝜑𝑀 ∈ ℤ) → (abs‘𝑀 / 𝑘𝐴) ∈ ℂ)
4729, 46eqeltrd 2254 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ)
48 prodsns 11606 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘(abs‘𝐴) ∈ ℂ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
4933, 47, 48syl2anc 411 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀} (abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5032, 49eqtrd 2210 . . . . 5 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴) = 𝑀 / 𝑘(abs‘𝐴))
5130prodeq1d 11567 . . . . . . . 8 (𝑀 ∈ ℤ → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
5251adantl 277 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = ∏𝑘 ∈ {𝑀}𝐴)
53 prodsns 11606 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑀 / 𝑘𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5433, 44, 53syl2anc 411 . . . . . . 7 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
5552, 54eqtrd 2210 . . . . . 6 ((𝜑𝑀 ∈ ℤ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = 𝑀 / 𝑘𝐴)
5655fveq2d 5519 . . . . 5 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = (abs‘𝑀 / 𝑘𝐴))
5729, 50, 563eqtr4rd 2221 . . . 4 ((𝜑𝑀 ∈ ℤ) → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴))
5857expcom 116 . . 3 (𝑀 ∈ ℤ → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑀)𝐴) = ∏𝑘 ∈ (𝑀...𝑀)(abs‘𝐴)))
59 simp3 999 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴))
60 peano2uz 9581 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
61 csbfv2g 5552 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (ℤ𝑀) → (𝑛 + 1) / 𝑘(abs‘𝐴) = (abs‘(𝑛 + 1) / 𝑘𝐴))
6260, 61syl 14 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) / 𝑘(abs‘𝐴) = (abs‘(𝑛 + 1) / 𝑘𝐴))
6362eqcomd 2183 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑀) → (abs‘(𝑛 + 1) / 𝑘𝐴) = (𝑛 + 1) / 𝑘(abs‘𝐴))
64633ad2ant2 1019 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀) ∧ (abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) = ∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴)) → (abs‘(𝑛 + 1) / 𝑘𝐴) = (𝑛 + 1) / 𝑘(abs‘𝐴))
6559, 64oveq12d 5892 . . . . . . 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 110 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ (ℤ𝑀))
67 elfzuz 10018 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘 ∈ (ℤ𝑀))
6867, 2eleqtrrdi 2271 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...(𝑛 + 1)) → 𝑘𝑍)
6968, 36sylan2 286 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7069adantlr 477 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → 𝐴 ∈ ℂ)
7166, 70fprodp1s 11605 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴))
7271fveq2d 5519 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)))
73 eluzel2 9531 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
7473adantl 277 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑀 ∈ ℤ)
75 eluzelz 9535 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
7675adantl 277 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℤ)
7774, 76fzfigd 10428 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑀...𝑛) ∈ Fin)
78 elfzuz 10018 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7978, 2eleqtrrdi 2271 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
8079, 36sylan2 286 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
8180adantlr 477 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐴 ∈ ℂ)
8277, 81fprodcl 11610 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...𝑛)𝐴 ∈ ℂ)
8360, 2eleqtrrdi 2271 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
84 nfcsb1v 3090 . . . . . . . . . . . . . 14 𝑘(𝑛 + 1) / 𝑘𝐴
8584nfel1 2330 . . . . . . . . . . . . 13 𝑘(𝑛 + 1) / 𝑘𝐴 ∈ ℂ
86 csbeq1a 3066 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → 𝐴 = (𝑛 + 1) / 𝑘𝐴)
8786eleq1d 2246 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8885, 87rspc 2835 . . . . . . . . . . . 12 ((𝑛 + 1) ∈ 𝑍 → (∀𝑘𝑍 𝐴 ∈ ℂ → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ))
8937, 88mpan9 281 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
9083, 89sylan2 286 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) / 𝑘𝐴 ∈ ℂ)
9182, 90absmuld 11198 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘(∏𝑘 ∈ (𝑀...𝑛)𝐴 · (𝑛 + 1) / 𝑘𝐴)) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
9272, 91eqtrd 2210 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (abs‘∏𝑘 ∈ (𝑀...(𝑛 + 1))𝐴) = ((abs‘∏𝑘 ∈ (𝑀...𝑛)𝐴) · (abs‘(𝑛 + 1) / 𝑘𝐴)))
93923adant3 1017 . . . . . . 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)))
9470abscld 11185 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℝ)
9594recnd 7984 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑀)) ∧ 𝑘 ∈ (𝑀...(𝑛 + 1))) → (abs‘𝐴) ∈ ℂ)
9666, 95fprodp1s 11605 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → ∏𝑘 ∈ (𝑀...(𝑛 + 1))(abs‘𝐴) = (∏𝑘 ∈ (𝑀...𝑛)(abs‘𝐴) · (𝑛 + 1) / 𝑘(abs‘𝐴)))
97963adant3 1017 . . . . . . 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)))
9865, 93, 973eqtr4d 2220 . . . . . 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))
99983exp 1202 . . . . 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))))
10099com12 30 . . . 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))))
101100a2d 26 . . 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))))
1029, 15, 21, 27, 58, 101uzind4 9586 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴)))
1033, 102mpcom 36 1 (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  csb 3057  {csn 3592  cfv 5216  (class class class)co 5874  cc 7808  1c1 7811   + caddc 7813   · cmul 7815  cz 9251  cuz 9526  ...cfz 10006  abscabs 11001  cprod 11553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-proddc 11554
This theorem is referenced by: (None)
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