Theorem ltmuls2d

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	ltmuls2d	|- ( ph -> ( A ( C x.s A )

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		ltmuls2	|- ( ( ( C e. No /\ 0s ( A ( C x.s A )
6	3 4 1 2 5	syl211anc	|- ( ph -> ( A ( C x.s A )