MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mu Structured version   Visualization version   GIF version

Definition df-mu 24572
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 24566 . 2 class μ
2 vx . . 3 setvar 𝑥
3 cn 10870 . . 3 class
4 vp . . . . . . . 8 setvar 𝑝
54cv 1473 . . . . . . 7 class 𝑝
6 c2 10920 . . . . . . 7 class 2
7 cexp 12680 . . . . . . 7 class
85, 6, 7co 6527 . . . . . 6 class (𝑝↑2)
92cv 1473 . . . . . 6 class 𝑥
10 cdvds 14770 . . . . . 6 class
118, 9, 10wbr 4577 . . . . 5 wff (𝑝↑2) ∥ 𝑥
12 cprime 15172 . . . . 5 class
1311, 4, 12wrex 2896 . . . 4 wff 𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥
14 cc0 9793 . . . 4 class 0
15 c1 9794 . . . . . 6 class 1
1615cneg 10119 . . . . 5 class -1
175, 9, 10wbr 4577 . . . . . . 7 wff 𝑝𝑥
1817, 4, 12crab 2899 . . . . . 6 class {𝑝 ∈ ℙ ∣ 𝑝𝑥}
19 chash 12937 . . . . . 6 class #
2018, 19cfv 5790 . . . . 5 class (#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})
2116, 20, 7co 6527 . . . 4 class (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))
2213, 14, 21cif 4035 . . 3 class if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})))
232, 3, 22cmpt 4637 . 2 class (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
241, 23wceq 1474 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  24603  muf  24611
  Copyright terms: Public domain W3C validator