Theorem logled

Description: Natural logarithm preserves <_ . (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	relogcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	relogmuld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	logled	$$⊢\phi \to \left(A\le B\leftrightarrow \log A\le \log B\right)$$

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		relogmuld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		logleb	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to \left(A\le B\leftrightarrow \log A\le \log B\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A\le B\leftrightarrow \log A\le \log B\right)$$