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Metamath Proof Explorer

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 )