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 [,] +∞ ) )