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 [,] +∞ ) )