Theorem cxpgt0d

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

	Ref	Expression
Hypotheses	cxpgt0d.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	cxpgt0d.2	$$⊢\phi \to N\in ℝ$$
Assertion	cxpgt0d	$$⊢\phi \to 0<A^N$$

Proof

Step	Hyp	Ref	Expression
1		cxpgt0d.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		cxpgt0d.2	$$⊢\phi \to N\in ℝ$$
3	1 2	rpcxpcld	$$⊢\phi \to A^N\in {ℝ}^{+}$$
4	3	rpgt0d	$$⊢\phi \to 0<A^N$$