Theorem pccl

Description: Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014)

	Ref	Expression
Assertion	pccl	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
2		nnne0	⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 )
3	1 2	jca	⊢ ( 𝑁 ∈ ℕ → ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) )
4		pczcl	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) ) → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )
5	3 4	sylan2	⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ( 𝑃 pCnt 𝑁 ) ∈ ℕ0 )