This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nnexpcld

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses nnexpcld.1 ⊢ φ → A ∈ ℕ
nnexpcld.2 ⊢ φ → N ∈ ℕ 0
Assertion nnexpcld ⊢ φ → A N ∈ ℕ

Proof

Step Hyp Ref Expression
1 nnexpcld.1 ⊢ φ → A ∈ ℕ
2 nnexpcld.2 ⊢ φ → N ∈ ℕ 0
3 nnexpcl ⊢ A ∈ ℕ ∧ N ∈ ℕ 0 → A N ∈ ℕ
4 1 2 3 syl2anc ⊢ φ → A N ∈ ℕ