Theorem pccld

Description: Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	pccld.1	⊢ ( 𝜑 → 𝑃 ∈ ℙ )
	pccld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	pccld	⊢ ( 𝜑 → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		pccld.1	⊢ ( 𝜑 → 𝑃 ∈ ℙ )
2		pccld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
3		pccl	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )