Theorem cxpgt0d

Description: A positive real raised to a real power is positive. (Contributed by SN, 6-Apr-2023)

	Ref	Expression
Hypotheses	cxpgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	cxpgt0d.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
Assertion	cxpgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 ↑𝑐 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		cxpgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		cxpgt0d.2	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
3	1 2	rpcxpcld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝑁 ) ∈ ℝ+ )
4	3	rpgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 ↑𝑐 𝑁 ) )