Theorem mulge0d

Description: The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	addge0d.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	addge0d.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion	mulge0d	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		addge0d.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
4		addge0d.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		mulge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → 0 ≤ ( 𝐴 · 𝐵 ) )