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Metamath Proof Explorer

Theorem lerec2d

Description: Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpred.1 ⊢ φ → A ∈ ℝ +
rpaddcld.1 ⊢ φ → B ∈ ℝ +
lerec2d.2 ⊢ φ → A ≤ 1 B
Assertion lerec2d ⊢ φ → B ≤ 1 A

Proof

Step Hyp Ref Expression
1 rpred.1 ⊢ φ → A ∈ ℝ +
2 rpaddcld.1 ⊢ φ → B ∈ ℝ +
3 lerec2d.2 ⊢ φ → A ≤ 1 B
4 1 rpregt0d ⊢ φ → A ∈ ℝ ∧ 0 < A
5 2 rpregt0d ⊢ φ → B ∈ ℝ ∧ 0 < B
6 lerec2 ⊢ A ∈ ℝ ∧ 0 < A ∧ B ∈ ℝ ∧ 0 < B → A ≤ 1 B ↔ B ≤ 1 A
7 4 5 6 syl2anc ⊢ φ → A ≤ 1 B ↔ B ≤ 1 A
8 3 7 mpbid ⊢ φ → B ≤ 1 A