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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 ∈ ℕ 0
Assertion zexpcld ⊢ φ → A N ∈ ℤ

Proof

Step Hyp Ref Expression
1 zexpcld.1 ⊢ φ → A ∈ ℤ
2 zexpcld.2 ⊢ φ → N ∈ ℕ 0
3 zexpcl ⊢ A ∈ ℤ ∧ N ∈ ℕ 0 → A N ∈ ℤ
4 1 2 3 syl2anc ⊢ φ → A N ∈ ℤ