Theorem addscomd

Description: Surreal addition is commutative. Part of Theorem 3 of Conway p. 17. (Contributed by Scott Fenton, 20-Aug-2024)

	Ref	Expression
Hypotheses	addscomd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	addscomd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	addscomd	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		addscomd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		addscomd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		addscom	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )