Theorem ledivdivd

Description: Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltdiv2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	ledivdivd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ+ )
	ledivdivd.5	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≤ ( 𝐶 / 𝐷 ) )
Assertion	ledivdivd	⊢ ( 𝜑 → ( 𝐷 / 𝐶 ) ≤ ( 𝐵 / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltdiv2d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		ledivdivd.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ+ )
5		ledivdivd.5	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≤ ( 𝐶 / 𝐷 ) )
6	1	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )
7	2	rpregt0d	⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
8	3	rpregt0d	⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) )
9	4	rpregt0d	⊢ ( 𝜑 → ( 𝐷 ∈ ℝ ∧ 0 < 𝐷 ) )
10		ledivdiv	⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) ∧ ( ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ∧ ( 𝐷 ∈ ℝ ∧ 0 < 𝐷 ) ) ) → ( ( 𝐴 / 𝐵 ) ≤ ( 𝐶 / 𝐷 ) ↔ ( 𝐷 / 𝐶 ) ≤ ( 𝐵 / 𝐴 ) ) )
11	6 7 8 9 10	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ≤ ( 𝐶 / 𝐷 ) ↔ ( 𝐷 / 𝐶 ) ≤ ( 𝐵 / 𝐴 ) ) )
12	5 11	mpbid	⊢ ( 𝜑 → ( 𝐷 / 𝐶 ) ≤ ( 𝐵 / 𝐴 ) )