Theorem cxpgt0d

Description: Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 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	relogcld	$$⊢\phi \to \log A\in ℝ$$
4	2 3	remulcld	$$⊢\phi \to N\log A\in ℝ$$
5		efgt0	$$⊢N\log A\in ℝ\to 0<{\mathrm{e}}^{N\log A}$$
6	4 5	syl	$$⊢\phi \to 0<{\mathrm{e}}^{N\log A}$$
7	1	rpcnd	$$⊢\phi \to A\in ℂ$$
8	1	rpne0d	$$⊢\phi \to A\ne 0$$
9	2	recnd	$$⊢\phi \to N\in ℂ$$
10	7 8 9	cxpefd	$$⊢\phi \to A^N={\mathrm{e}}^{N\log A}$$
11	6 10	breqtrrd	$$⊢\phi \to 0<A^N$$