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Metamath Proof Explorer

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 ) )