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Metamath Proof Explorer

Theorem lediv1dd

Description: Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses ltmul1d.1 ⊢ φ → A ∈ ℝ
ltmul1d.2 ⊢ φ → B ∈ ℝ
ltmul1d.3 ⊢ φ → C ∈ ℝ +
lediv1dd.4 ⊢ φ → A ≤ B
Assertion lediv1dd ⊢ φ → A C ≤ 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 lediv1dd.4 ⊢ φ → A ≤ B
5 1 2 3 lediv1d ⊢ φ → A ≤ B ↔ A C ≤ B C
6 4 5 mpbid ⊢ φ → A C ≤ B C