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Metamath Proof Explorer

Theorem zexpcld

Description: Closure of exponentiation of integers, deduction form. (Contributed by SN, 15-Sep-2024)

Ref Expression
Hypotheses zexpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℤ )
zexpcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion zexpcld ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℤ )

Proof

Step Hyp Ref Expression
1 zexpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℤ )
2 zexpcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3 zexpcl ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℤ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℤ )