Theorem ltmuldivswd

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	ltmuldivswd	|- ( ph -> ( ( A x.s 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	4	gt0ne0sd	|- ( ph -> C =/= 0s )
7	2 3 6 5	divsclwd	|- ( ph -> ( B /su C ) e. No )
8	1 7 3 4	ltmuls1d	|- ( ph -> ( A ( A x.s C )
9	2 3 6 5	divscan1wd	|- ( ph -> ( ( B /su C ) x.s C ) = B )
10	9	breq2d	|- ( ph -> ( ( A x.s C ) ( A x.s C )
11	8 10	bitr2d	|- ( ph -> ( ( A x.s C ) A