Theorem mulsassd

Description: Associative law for surreal multiplication. Part of theorem 7 of Conway p. 19. (Contributed by Scott Fenton, 10-Mar-2025)

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

Proof

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