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Metamath Proof Explorer

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 [,] +∞ ) )