Theorem ltexp2rd

Description: The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpexpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpexpcld.2	$$⊢\phi \to N\in \mathbb{Z}$$
	ltexp2rd.3	$$⊢\phi \to M\in \mathbb{Z}$$
	ltexp2rd.4	$$⊢\phi \to A<1$$
Assertion	ltexp2rd	$$⊢\phi \to \left(M<N\leftrightarrow A^N<A^M\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpexpcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpexpcld.2	$$⊢\phi \to N\in \mathbb{Z}$$
3		ltexp2rd.3	$$⊢\phi \to M\in \mathbb{Z}$$
4		ltexp2rd.4	$$⊢\phi \to A<1$$
5		ltexp2r	$$⊢\left(\left(A\in {ℝ}^{+}\wedge M\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\wedge A<1\right)\to \left(M<N\leftrightarrow A^N<A^M\right)$$
6	1 3 2 4 5	syl31anc	$$⊢\phi \to \left(M<N\leftrightarrow A^N<A^M\right)$$