Theorem muval2

Description: The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014)

	Ref	Expression
Assertion	muval2	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ) → ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ne	⊢ ( ( μ ‘ 𝐴 ) ≠ 0 ↔ ¬ ( μ ‘ 𝐴 ) = 0 )
2		ifeqor	⊢ ( if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = 0 ∨ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) )
3		muval	⊢ ( 𝐴 ∈ ℕ → ( μ ‘ 𝐴 ) = if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) )
4	3	eqeq1d	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) = 0 ↔ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = 0 ) )
5	3	eqeq1d	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ↔ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) )
6	4 5	orbi12d	⊢ ( 𝐴 ∈ ℕ → ( ( ( μ ‘ 𝐴 ) = 0 ∨ ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) ↔ ( if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = 0 ∨ if ( ∃ 𝑝 ∈ ℙ ( 𝑝 ↑ 2 ) ∥ 𝐴 , 0 , ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) ) )
7	2 6	mpbiri	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) = 0 ∨ ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) )
8	7	ord	⊢ ( 𝐴 ∈ ℕ → ( ¬ ( μ ‘ 𝐴 ) = 0 → ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) )
9	1 8	biimtrid	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) ≠ 0 → ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) ) )
10	9	imp	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ) → ( μ ‘ 𝐴 ) = ( - 1 ↑ ( ♯ ‘ { 𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴 } ) ) )