Theorem ltmuls1d

Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	ltmuls12d.1	|- ( ph -> A e. No )
	ltmuls12d.2	|- ( ph -> B e. No )
	ltmuls12d.3	|- ( ph -> C e. No )
	ltmuls12d.4	|- ( ph -> 0s
Assertion	ltmuls1d	|- ( ph -> ( A ( A x.s C )

Proof

Step	Hyp	Ref	Expression
1		ltmuls12d.1	|- ( ph -> A e. No )
2		ltmuls12d.2	|- ( ph -> B e. No )
3		ltmuls12d.3	|- ( ph -> C e. No )
4		ltmuls12d.4	|- ( ph -> 0s
5	1 2 3 4	ltmuls2d	|- ( ph -> ( A ( C x.s A )
6	1 3	mulscomd	|- ( ph -> ( A x.s C ) = ( C x.s A ) )
7	2 3	mulscomd	|- ( ph -> ( B x.s C ) = ( C x.s B ) )
8	6 7	breq12d	|- ( ph -> ( ( A x.s C ) ( C x.s A )
9	5 8	bitr4d	|- ( ph -> ( A ( A x.s C )