Theorem ltmulnegs2d

Description: Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	ltmulnegs.1	|- ( ph -> A e. No )
	ltmulnegs.2	|- ( ph -> B e. No )
	ltmulnegs.3	|- ( ph -> C e. No )
	ltmulnegs.4	|- ( ph -> C
Assertion	ltmulnegs2d	|- ( ph -> ( A ( C x.s B )

Proof

Step	Hyp	Ref	Expression
1		ltmulnegs.1	|- ( ph -> A e. No )
2		ltmulnegs.2	|- ( ph -> B e. No )
3		ltmulnegs.3	|- ( ph -> C e. No )
4		ltmulnegs.4	|- ( ph -> C
5	1 2 3 4	ltmulnegs1d	|- ( ph -> ( A ( B x.s C )
6	2 3	mulscomd	|- ( ph -> ( B x.s C ) = ( C x.s B ) )
7	1 3	mulscomd	|- ( ph -> ( A x.s C ) = ( C x.s A ) )
8	6 7	breq12d	|- ( ph -> ( ( B x.s C ) ( C x.s B )
9	5 8	bitrd	|- ( ph -> ( A ( C x.s B )