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

Metamath Proof Explorer

Theorem nn0expcld

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

Ref Expression
Hypotheses nn0expcld.1 ⊢ φ → A ∈ ℕ 0
nn0expcld.2 ⊢ φ → N ∈ ℕ 0
Assertion nn0expcld ⊢ φ → A N ∈ ℕ 0

Proof

Step Hyp Ref Expression
1 nn0expcld.1 ⊢ φ → A ∈ ℕ 0
2 nn0expcld.2 ⊢ φ → N ∈ ℕ 0
3 nn0expcl ⊢ A ∈ ℕ 0 ∧ N ∈ ℕ 0 → A N ∈ ℕ 0
4 1 2 3 syl2anc ⊢ φ → A N ∈ ℕ 0