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Metamath Proof Explorer

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 ) )