Theorem xaddcomd

Description: The extended real addition operation is commutative. (Contributed by Glauco Siliprandi, 17-Aug-2020)

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

Proof

Step	Hyp	Ref	Expression
1		xaddcomd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xaddcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xaddcom	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 +𝑒 𝐵 ) = ( 𝐵 +𝑒 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +𝑒 𝐵 ) = ( 𝐵 +𝑒 𝐴 ) )