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Metamath Proof Explorer

Theorem expge0d

Description: A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses reexpcld.1 ⊢ φ → A ∈ ℝ
reexpcld.2 ⊢ φ → N ∈ ℕ 0
expge0d.3 ⊢ φ → 0 ≤ A
Assertion expge0d ⊢ φ → 0 ≤ A N

Proof

Step Hyp Ref Expression
1 reexpcld.1 ⊢ φ → A ∈ ℝ
2 reexpcld.2 ⊢ φ → N ∈ ℕ 0
3 expge0d.3 ⊢ φ → 0 ≤ A
4 expge0 ⊢ A ∈ ℝ ∧ N ∈ ℕ 0 ∧ 0 ≤ A → 0 ≤ A N
5 1 2 3 4 syl3anc ⊢ φ → 0 ≤ A N