Theorem lerecd

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	lerecd	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3	1	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
4	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
5		lerec	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ) )
6	3 4 5	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 1 / 𝐵 ) ≤ ( 1 / 𝐴 ) ) )