Theorem mulscan1d

Description: Cancellation of surreal multiplication when the left term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	mulscan2d.1	|- ( ph -> A e. No )
	mulscan2d.2	|- ( ph -> B e. No )
	mulscan2d.3	|- ( ph -> C e. No )
	mulscan2d.4	|- ( ph -> C =/= 0s )
Assertion	mulscan1d	|- ( ph -> ( ( C x.s A ) = ( C x.s B ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		mulscan2d.1	|- ( ph -> A e. No )
2		mulscan2d.2	|- ( ph -> B e. No )
3		mulscan2d.3	|- ( ph -> C e. No )
4		mulscan2d.4	|- ( ph -> C =/= 0s )
5	1 3	mulscomd	|- ( ph -> ( A x.s C ) = ( C x.s A ) )
6	2 3	mulscomd	|- ( ph -> ( B x.s C ) = ( C x.s B ) )
7	5 6	eqeq12d	|- ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> ( C x.s A ) = ( C x.s B ) ) )
8	1 2 3 4	mulscan2d	|- ( ph -> ( ( A x.s C ) = ( B x.s C ) <-> A = B ) )
9	7 8	bitr3d	|- ( ph -> ( ( C x.s A ) = ( C x.s B ) <-> A = B ) )