Theorem leexp2d

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

	Ref	Expression
Hypotheses	sqgt0d.1	$$⊢\phi \to A\in ℝ$$
	ltexp2d.2	$$⊢\phi \to M\in \mathbb{Z}$$
	ltexp2d.3	$$⊢\phi \to N\in \mathbb{Z}$$
	ltexp2d.4	$$⊢\phi \to 1<A$$
Assertion	leexp2d	$$⊢\phi \to \left(M\le N\leftrightarrow A^M\le A^N\right)$$

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	$$⊢\phi \to A\in ℝ$$
2		ltexp2d.2	$$⊢\phi \to M\in \mathbb{Z}$$
3		ltexp2d.3	$$⊢\phi \to N\in \mathbb{Z}$$
4		ltexp2d.4	$$⊢\phi \to 1<A$$
5		leexp2	$$⊢\left(\left(A\in ℝ\wedge M\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\wedge 1<A\right)\to \left(M\le N\leftrightarrow A^M\le A^N\right)$$
6	1 2 3 4 5	syl31anc	$$⊢\phi \to \left(M\le N\leftrightarrow A^M\le A^N\right)$$