Theorem lemul1ad

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	lemul1ad	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) )

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		lemul1a	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 ≤ 𝐶 ) ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) )
8	1 2 6 5 7	syl31anc	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ≤ ( 𝐵 · 𝐶 ) )