Theorem divscan1wd

Description: A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025)

	Ref	Expression
Hypotheses	divscan2wd.1	|- ( ph -> A e. No )
	divscan2wd.2	|- ( ph -> B e. No )
	divscan2wd.3	|- ( ph -> B =/= 0s )
	divscan2wd.4	|- ( ph -> E. x e. No ( B x.s x ) = 1s )
Assertion	divscan1wd	|- ( ph -> ( ( A /su B ) x.s B ) = A )

Proof

Step	Hyp	Ref	Expression
1		divscan2wd.1	|- ( ph -> A e. No )
2		divscan2wd.2	|- ( ph -> B e. No )
3		divscan2wd.3	|- ( ph -> B =/= 0s )
4		divscan2wd.4	|- ( ph -> E. x e. No ( B x.s x ) = 1s )
5	1 2 3 4	divsclwd	|- ( ph -> ( A /su B ) e. No )
6	5 2	mulscomd	|- ( ph -> ( ( A /su B ) x.s B ) = ( B x.s ( A /su B ) ) )
7	1 2 3 4	divscan2wd	|- ( ph -> ( B x.s ( A /su B ) ) = A )
8	6 7	eqtrd	|- ( ph -> ( ( A /su B ) x.s B ) = A )