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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 ) )