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Metamath Proof Explorer

Theorem nn0expcli

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015)

Ref Expression
Hypotheses nn0expcli.1 ⊢ A ∈ ℕ 0
nn0expcli.2 ⊢ N ∈ ℕ 0
Assertion nn0expcli ⊢ A N ∈ ℕ 0

Proof

Step Hyp Ref Expression
1 nn0expcli.1 ⊢ A ∈ ℕ 0
2 nn0expcli.2 ⊢ N ∈ ℕ 0
3 nn0expcl ⊢ A ∈ ℕ 0 ∧ N ∈ ℕ 0 → A N ∈ ℕ 0
4 1 2 3 mp2an ⊢ A N ∈ ℕ 0