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Metamath Proof Explorer

Theorem xrge0addcld

Description: Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019)

Ref Expression
Hypotheses xrge0addcld.a ⊢ ( 𝜑 → 𝐴 ∈ ( 0 [,] +∞ ) )
xrge0addcld.b ⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) )
Assertion xrge0addcld ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) )

Proof

Step Hyp Ref Expression
1 xrge0addcld.a ⊢ ( 𝜑 → 𝐴 ∈ ( 0 [,] +∞ ) )
2 xrge0addcld.b ⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) )
3 elxrge0 ⊢ ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) )
4 1 3 sylib ⊢ ( 𝜑 → ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) )
5 4 simpld ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
6 elxrge0 ⊢ ( 𝐵 ∈ ( 0 [,] +∞ ) ↔ ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) )
7 2 6 sylib ⊢ ( 𝜑 → ( 𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵 ) )
8 7 simpld ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
9 5 8 xaddcld ⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )
10 4 simprd ⊢ ( 𝜑 → 0 ≤ 𝐴 )
11 7 simprd ⊢ ( 𝜑 → 0 ≤ 𝐵 )
12 xaddge0 ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 +𝑒 𝐵 ) )
13 5 8 10 11 12 syl22anc ⊢ ( 𝜑 → 0 ≤ ( 𝐴 +𝑒 𝐵 ) )
14 elxrge0 ⊢ ( ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐴 +𝑒 𝐵 ) ) )
15 9 13 14 sylanbrc ⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) ∈ ( 0 [,] +∞ ) )