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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