Theorem divsmul

Description: Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Assertion	divsmul	|- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		recsex	|- ( ( C e. No /\ C =/= 0s ) -> E. x e. No ( C x.s x ) = 1s )
2	1	3ad2ant3	|- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> E. x e. No ( C x.s x ) = 1s )
3		divsmulw	|- ( ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) /\ E. x e. No ( C x.s x ) = 1s ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) )
4	2 3	mpdan	|- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) )