Theorem ge0mulcl

Description: The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Assertion	ge0mulcl	⊢ ( ( 𝐴 ∈ ( 0 [,) +∞ ) ∧ 𝐵 ∈ ( 0 [,) +∞ ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		elrege0	⊢ ( 𝐴 ∈ ( 0 [,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
2		elrege0	⊢ ( 𝐵 ∈ ( 0 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) )
3		remulcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
4	3	ad2ant2r	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
5		mulge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) )
6		elrege0	⊢ ( ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐴 · 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 · 𝐵 ) ) )
7	4 5 6	sylanbrc	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )
8	1 2 7	syl2anb	⊢ ( ( 𝐴 ∈ ( 0 [,) +∞ ) ∧ 𝐵 ∈ ( 0 [,) +∞ ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )