This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
divgt0d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
lemul1ad.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
lemul1ad.4 ⊢ ( 𝜑 → 0 ≤ 𝐶 )
lemul1ad.5 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion lemul2ad ( 𝜑 → ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) )

Proof

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