Theorem leexp2ad

Description: Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	$$⊢\phi \to A\in ℝ$$
	leexp2ad.2	$$⊢\phi \to 1\le A$$
	leexp2ad.3	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
Assertion	leexp2ad	$$⊢\phi \to A^M\le A^N$$

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	$$⊢\phi \to A\in ℝ$$
2		leexp2ad.2	$$⊢\phi \to 1\le A$$
3		leexp2ad.3	$$⊢\phi \to N\in {\mathbb{Z}}_{\ge M}$$
4		leexp2a	$$⊢\left(A\in ℝ\wedge 1\le A\wedge N\in {\mathbb{Z}}_{\ge M}\right)\to A^M\le A^N$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A^M\le A^N$$