Theorem ge0xmulcl

Description: The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Assertion	ge0xmulcl	⊢ ( ( 𝐴 ∈ ( 0 [,] +∞ ) ∧ 𝐵 ∈ ( 0 [,] +∞ ) ) → ( 𝐴 ·e 𝐵 ) ∈ ( 0 [,] +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		elxrge0	⊢ ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) )
2		elxrge0	⊢ ( 𝐵 ∈ ( 0 [,] +∞ ) ↔ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) )
3		xmulcl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ·e 𝐵 ) ∈ ℝ* )
4	3	ad2ant2r	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ·e 𝐵 ) ∈ ℝ* )
5		xmulge0	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 ·e 𝐵 ) )
6		elxrge0	⊢ ( ( 𝐴 ·e 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐴 ·e 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐴 ·e 𝐵 ) ) )
7	4 5 6	sylanbrc	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ·e 𝐵 ) ∈ ( 0 [,] +∞ ) )
8	1 2 7	syl2anb	⊢ ( ( 𝐴 ∈ ( 0 [,] +∞ ) ∧ 𝐵 ∈ ( 0 [,] +∞ ) ) → ( 𝐴 ·e 𝐵 ) ∈ ( 0 [,] +∞ ) )