Theorem slemul2d

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

	Ref	Expression
Hypotheses	sltmul12d.1	|- ( ph -> A e. No )
	sltmul12d.2	|- ( ph -> B e. No )
	sltmul12d.3	|- ( ph -> C e. No )
	sltmul12d.4	|- ( ph -> 0s
Assertion	slemul2d	|- ( ph -> ( A <_s B <-> ( C x.s A ) <_s ( C x.s B ) ) )

Proof

Step	Hyp	Ref	Expression
1		sltmul12d.1	|- ( ph -> A e. No )
2		sltmul12d.2	|- ( ph -> B e. No )
3		sltmul12d.3	|- ( ph -> C e. No )
4		sltmul12d.4	|- ( ph -> 0s
5	2 1 3 4	sltmul2d	|- ( ph -> ( B ( C x.s B )
6	5	notbid	|- ( ph -> ( -. B -. ( C x.s B )
7		slenlt	|- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> -. B
8	1 2 7	syl2anc	|- ( ph -> ( A <_s B <-> -. B
9	3 1	mulscld	|- ( ph -> ( C x.s A ) e. No )
10	3 2	mulscld	|- ( ph -> ( C x.s B ) e. No )
11		slenlt	|- ( ( ( C x.s A ) e. No /\ ( C x.s B ) e. No ) -> ( ( C x.s A ) <_s ( C x.s B ) <-> -. ( C x.s B )
12	9 10 11	syl2anc	|- ( ph -> ( ( C x.s A ) <_s ( C x.s B ) <-> -. ( C x.s B )
13	6 8 12	3bitr4d	|- ( ph -> ( A <_s B <-> ( C x.s A ) <_s ( C x.s B ) ) )