Theorem lediv12ad

Description: Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	lediv12ad.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	lediv12ad.5	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	lediv12ad.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	lediv12ad.7	⊢ ( 𝜑 → 𝐶 ≤ 𝐷 )
Assertion	lediv12ad	⊢ ( 𝜑 → ( 𝐴 / 𝐷 ) ≤ ( 𝐵 / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ltmul1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltmul1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltmul1d.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		lediv12ad.4	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
5		lediv12ad.5	⊢ ( 𝜑 → 0 ≤ 𝐴 )
6		lediv12ad.6	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
7		lediv12ad.7	⊢ ( 𝜑 → 𝐶 ≤ 𝐷 )
8	1 2	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
9	5 6	jca	⊢ ( 𝜑 → ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) )
10	3	rpred	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
11	10 4	jca	⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) )
12	3	rpgt0d	⊢ ( 𝜑 → 0 < 𝐶 )
13	12 7	jca	⊢ ( 𝜑 → ( 0 < 𝐶 ∧ 𝐶 ≤ 𝐷 ) )
14		lediv12a	⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ ( ( 𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ) ∧ ( 0 < 𝐶 ∧ 𝐶 ≤ 𝐷 ) ) ) → ( 𝐴 / 𝐷 ) ≤ ( 𝐵 / 𝐶 ) )
15	8 9 11 13 14	syl22anc	⊢ ( 𝜑 → ( 𝐴 / 𝐷 ) ≤ ( 𝐵 / 𝐶 ) )