Theorem ltmuldivs2wd

Description: Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	ltdivmulswd.1	|- ( ph -> A e. No )
	ltdivmulswd.2	|- ( ph -> B e. No )
	ltdivmulswd.3	|- ( ph -> C e. No )
	ltdivmulswd.4	|- ( ph -> 0s
	ltdivmulswd.5	|- ( ph -> E. x e. No ( C x.s x ) = 1s )
Assertion	ltmuldivs2wd	|- ( ph -> ( ( C x.s A ) A

Proof

Step	Hyp	Ref	Expression
1		ltdivmulswd.1	|- ( ph -> A e. No )
2		ltdivmulswd.2	|- ( ph -> B e. No )
3		ltdivmulswd.3	|- ( ph -> C e. No )
4		ltdivmulswd.4	|- ( ph -> 0s
5		ltdivmulswd.5	|- ( ph -> E. x e. No ( C x.s x ) = 1s )
6	1 3	mulscomd	|- ( ph -> ( A x.s C ) = ( C x.s A ) )
7	6	breq1d	|- ( ph -> ( ( A x.s C ) ( C x.s A )
8	1 2 3 4 5	ltmuldivswd	|- ( ph -> ( ( A x.s C ) A
9	7 8	bitr3d	|- ( ph -> ( ( C x.s A ) A