Theorem divsclwd

Description: Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		divsclwd.1	|- ( ph -> A e. No )
2		divsclwd.2	|- ( ph -> B e. No )
3		divsclwd.3	|- ( ph -> B =/= 0s )
4		divsclwd.4	|- ( ph -> E. x e. No ( B x.s x ) = 1s )
5		divsclw	|- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) e. No )
6	1 2 3 4 5	syl31anc	|- ( ph -> ( A /su B ) e. No )