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Definition df-mu 25686
Description: Define the Möbius function, which is zero for non-squarefree numbers and is -1 or 1 for squarefree numbers according as to the number of prime divisors of the number is even or odd, see definition in [ApostolNT] p. 24. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
df-mu μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
Distinct variable group:   𝑥,𝑝

Detailed syntax breakdown of Definition df-mu
StepHypRef Expression
1 cmu 25680 . 2 class μ
2 vx . . 3 setvar 𝑥
3 cn 11625 . . 3 class
4 vp . . . . . . . 8 setvar 𝑝
54cv 1537 . . . . . . 7 class 𝑝
6 c2 11680 . . . . . . 7 class 2
7 cexp 13425 . . . . . . 7 class
85, 6, 7co 7135 . . . . . 6 class (𝑝↑2)
92cv 1537 . . . . . 6 class 𝑥
10 cdvds 15599 . . . . . 6 class
118, 9, 10wbr 5030 . . . . 5 wff (𝑝↑2) ∥ 𝑥
12 cprime 16005 . . . . 5 class
1311, 4, 12wrex 3107 . . . 4 wff 𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥
14 cc0 10526 . . . 4 class 0
15 c1 10527 . . . . . 6 class 1
1615cneg 10860 . . . . 5 class -1
175, 9, 10wbr 5030 . . . . . . 7 wff 𝑝𝑥
1817, 4, 12crab 3110 . . . . . 6 class {𝑝 ∈ ℙ ∣ 𝑝𝑥}
19 chash 13686 . . . . . 6 class
2018, 19cfv 6324 . . . . 5 class (♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})
2116, 20, 7co 7135 . . . 4 class (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))
2213, 14, 21cif 4425 . . 3 class if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})))
232, 3, 22cmpt 5110 . 2 class (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
241, 23wceq 1538 1 wff μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
Colors of variables: wff setvar class
This definition is referenced by:  muval  25717  muf  25725
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