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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 ∈ ℕ0 )
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 ∈ ℕ0 )
3 expge0d.3 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
4 expge0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑁 ) )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → 0 ≤ ( 𝐴 ↑ 𝑁 ) )