Theorem mulge0i

Description: The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	mulge0i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → 0 ≤ ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		mulge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) )
4	3	an4s	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) )
5	1 2 4	mpanl12	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → 0 ≤ ( 𝐴 · 𝐵 ) )