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 ) |
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 ) |