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Metamath Proof Explorer

Theorem ge0xaddcl

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

Ref Expression
Assertion ge0xaddcl ( ( 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 xaddcl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )
4 3 ad2ant2r ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )
5 xaddge0 ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 +𝑒 𝐵 ) )
6 5 an4s ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 +𝑒 𝐵 ) )
7 elxrge0 ⊢ ( ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐴 +𝑒 𝐵 ) ) )
8 4 6 7 sylanbrc ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) )
9 1 2 8 syl2anb ⊢ ( ( 𝐴 ∈ ( 0 [,] +∞ ) ∧ 𝐵 ∈ ( 0 [,] +∞ ) ) → ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) )