Theorem slemul1d

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	slemul1d	|- ( ph -> ( A <_s B <-> ( A x.s C ) <_s ( B x.s C ) ) )

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	sltmul1d	|- ( ph -> ( B ( B x.s C )
6	5	notbid	|- ( ph -> ( -. B -. ( B x.s C )
7		slenlt	|- ( ( A e. No /\ B e. No ) -> ( A <_s B <-> -. B
8	1 2 7	syl2anc	|- ( ph -> ( A <_s B <-> -. B
9	1 3	mulscld	|- ( ph -> ( A x.s C ) e. No )
10	2 3	mulscld	|- ( ph -> ( B x.s C ) e. No )
11		slenlt	|- ( ( ( A x.s C ) e. No /\ ( B x.s C ) e. No ) -> ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C )
12	9 10 11	syl2anc	|- ( ph -> ( ( A x.s C ) <_s ( B x.s C ) <-> -. ( B x.s C )
13	6 8 12	3bitr4d	|- ( ph -> ( A <_s B <-> ( A x.s C ) <_s ( B x.s C ) ) )