Theorem xaddcld

Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	xnegcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xaddcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
Assertion	xaddcld	⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		xnegcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xaddcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xaddcl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) ∈ ℝ* )