Theorem sqfpc

Description: The prime count of a squarefree number is at most 1. (Contributed by Mario Carneiro, 1-Jul-2015)

	Ref	Expression
Assertion	sqfpc	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ∧ 𝑃 ∈ ℙ ) → ( 𝑃 pCnt 𝐴 ) ≤ 1 )

Proof

Step	Hyp	Ref	Expression
1		issqf	⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) ≠ 0 ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ) )
2	1	biimpa	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ) → ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 )
3		oveq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 pCnt 𝐴 ) = ( 𝑃 pCnt 𝐴 ) )
4	3	breq1d	⊢ ( 𝑝 = 𝑃 → ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ↔ ( 𝑃 pCnt 𝐴 ) ≤ 1 ) )
5	4	rspccv	⊢ ( ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 → ( 𝑃 ∈ ℙ → ( 𝑃 pCnt 𝐴 ) ≤ 1 ) )
6	2 5	syl	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ) → ( 𝑃 ∈ ℙ → ( 𝑃 pCnt 𝐴 ) ≤ 1 ) )
7	6	3impia	⊢ ( ( 𝐴 ∈ ℕ ∧ ( μ ‘ 𝐴 ) ≠ 0 ∧ 𝑃 ∈ ℙ ) → ( 𝑃 pCnt 𝐴 ) ≤ 1 )