This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )