Theorem cxpge0d

Description: Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	cxpge0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 ↑𝑐 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		recxpcld.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4		cxpge0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ ) → 0 ≤ ( 𝐴 ↑𝑐 𝐵 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → 0 ≤ ( 𝐴 ↑𝑐 𝐵 ) )