Theorem divscl

Description: Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Assertion	divscl	|- ( ( A e. No /\ B e. No /\ B =/= 0s ) -> ( A /su B ) e. No )

Proof

Step	Hyp	Ref	Expression
1		recsex	|- ( ( B e. No /\ B =/= 0s ) -> E. x e. No ( B x.s x ) = 1s )
2	1	3adant1	|- ( ( A e. No /\ B e. No /\ B =/= 0s ) -> E. x e. No ( B x.s x ) = 1s )
3		divsclw	|- ( ( ( A e. No /\ B e. No /\ B =/= 0s ) /\ E. x e. No ( B x.s x ) = 1s ) -> ( A /su B ) e. No )
4	2 3	mpdan	|- ( ( A e. No /\ B e. No /\ B =/= 0s ) -> ( A /su B ) e. No )