Theorem logled

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

	Ref	Expression
Hypotheses	relogcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	relogmuld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	logled	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( log ‘ 𝐴 ) ≤ ( log ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		relogmuld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		logleb	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 ≤ 𝐵 ↔ ( log ‘ 𝐴 ) ≤ ( log ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( log ‘ 𝐴 ) ≤ ( log ‘ 𝐵 ) ) )