Theorem zexpcld

Description: Closure of exponentiation of integers, deductive 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 )