Theorem addsassd

Description: Surreal addition is associative. Part of theorem 3 of Conway p. 17. (Contributed by Scott Fenton, 22-Jan-2025)

	Ref	Expression
Hypotheses	addsassd.1	|- ( ph -> A e. No )
	addsassd.2	|- ( ph -> B e. No )
	addsassd.3	|- ( ph -> C e. No )
Assertion	addsassd	|- ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	|- ( ph -> A e. No )
2		addsassd.2	|- ( ph -> B e. No )
3		addsassd.3	|- ( ph -> C e. No )
4		addsass	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( ( A +s B ) +s C ) = ( A +s ( B +s C ) ) )