Theorem ltmulnegs1d

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	ltmulnegs1d	|- ( ph -> ( A ( B x.s C )

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 3	mulnegs2d	|- ( ph -> ( A x.s ( -us ` C ) ) = ( -us ` ( A x.s C ) ) )
6	2 3	mulnegs2d	|- ( ph -> ( B x.s ( -us ` C ) ) = ( -us ` ( B x.s C ) ) )
7	5 6	breq12d	|- ( ph -> ( ( A x.s ( -us ` C ) ) ( -us ` ( A x.s C ) )
8	3	negscld	|- ( ph -> ( -us ` C ) e. No )
9		neg0s	|- ( -us ` 0s ) = 0s
10		0no	|- 0s e. No
11	10	a1i	|- ( ph -> 0s e. No )
12	3 11	ltnegsd	|- ( ph -> ( C ( -us ` 0s )
13	4 12	mpbid	|- ( ph -> ( -us ` 0s )
14	9 13	eqbrtrrid	|- ( ph -> 0s
15	1 2 8 14	ltmuls1d	|- ( ph -> ( A ( A x.s ( -us ` C ) )
16	2 3	mulscld	|- ( ph -> ( B x.s C ) e. No )
17	1 3	mulscld	|- ( ph -> ( A x.s C ) e. No )
18	16 17	ltnegsd	|- ( ph -> ( ( B x.s C ) ( -us ` ( A x.s C ) )
19	7 15 18	3bitr4d	|- ( ph -> ( A ( B x.s C )