Theorem ltdivmuls2wd

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	ltdivmuls2wd	|- ( ph -> ( ( A /su C ) 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 2 3 4 5	ltdivmulswd	|- ( ph -> ( ( A /su C ) A
7	2 3	mulscomd	|- ( ph -> ( B x.s C ) = ( C x.s B ) )
8	7	breq2d	|- ( ph -> ( A A
9	6 8	bitr4d	|- ( ph -> ( ( A /su C ) A