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Metamath Proof Explorer

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 ) )