Theorem muls12d

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

	Ref	Expression
Hypotheses	muls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
	muls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
	muls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	muls12d	Could not format assertion : No typesetting found for |- ( ph -> ( A x.s ( B x.s C ) ) = ( B x.s ( A x.s C ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		muls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		muls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		muls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
4	1 2	mulscomd	Could not format ( ph -> ( A x.s B ) = ( B x.s A ) ) : No typesetting found for |- ( ph -> ( A x.s B ) = ( B x.s A ) ) with typecode |-
5	4	oveq1d	Could not format ( ph -> ( ( A x.s B ) x.s C ) = ( ( B x.s A ) x.s C ) ) : No typesetting found for |- ( ph -> ( ( A x.s B ) x.s C ) = ( ( B x.s A ) x.s C ) ) with typecode |-
6	1 2 3	mulsassd	Could not format ( ph -> ( ( A x.s B ) x.s C ) = ( A x.s ( B x.s C ) ) ) : No typesetting found for |- ( ph -> ( ( A x.s B ) x.s C ) = ( A x.s ( B x.s C ) ) ) with typecode |-
7	2 1 3	mulsassd	Could not format ( ph -> ( ( B x.s A ) x.s C ) = ( B x.s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( ( B x.s A ) x.s C ) = ( B x.s ( A x.s C ) ) ) with typecode |-
8	5 6 7	3eqtr3d	Could not format ( ph -> ( A x.s ( B x.s C ) ) = ( B x.s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> ( A x.s ( B x.s C ) ) = ( B x.s ( A x.s C ) ) ) with typecode |-