Metamath Proof Explorer


Theorem lemul2ad

Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses ltp1d.1 ( 𝜑𝐴 ∈ ℝ )
divgt0d.2 ( 𝜑𝐵 ∈ ℝ )
lemul1ad.3 ( 𝜑𝐶 ∈ ℝ )
lemul1ad.4 ( 𝜑 → 0 ≤ 𝐶 )
lemul1ad.5 ( 𝜑𝐴𝐵 )
Assertion lemul2ad ( 𝜑 → ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltp1d.1 ( 𝜑𝐴 ∈ ℝ )
2 divgt0d.2 ( 𝜑𝐵 ∈ ℝ )
3 lemul1ad.3 ( 𝜑𝐶 ∈ ℝ )
4 lemul1ad.4 ( 𝜑 → 0 ≤ 𝐶 )
5 lemul1ad.5 ( 𝜑𝐴𝐵 )
6 3 4 jca ( 𝜑 → ( 𝐶 ∈ ℝ ∧ 0 ≤ 𝐶 ) )
7 lemul2a ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴𝐵 ) → ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) )
8 1 2 6 5 7 syl31anc ( 𝜑 → ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) )