Theorem muls12d

Description: Commutative/associative law for surreal multiplication. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	muls12d.1	|- ( ph -> A e. No )
	muls12d.2	|- ( ph -> B e. No )
	muls12d.3	|- ( ph -> C e. No )
Assertion	muls12d	|- ( ph -> ( A x.s ( B x.s C ) ) = ( B x.s ( A x.s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		muls12d.1	|- ( ph -> A e. No )
2		muls12d.2	|- ( ph -> B e. No )
3		muls12d.3	|- ( ph -> C e. No )
4	1 2	mulscomd	|- ( ph -> ( A x.s B ) = ( B x.s A ) )
5	4	oveq1d	|- ( ph -> ( ( A x.s B ) x.s C ) = ( ( B x.s A ) x.s C ) )
6	1 2 3	mulsassd	|- ( ph -> ( ( A x.s B ) x.s C ) = ( A x.s ( B x.s C ) ) )
7	2 1 3	mulsassd	|- ( ph -> ( ( B x.s A ) x.s C ) = ( B x.s ( A x.s C ) ) )
8	5 6 7	3eqtr3d	|- ( ph -> ( A x.s ( B x.s C ) ) = ( B x.s ( A x.s C ) ) )