Theorem divsmul

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

	Ref	Expression
Assertion	divsmul	Could not format assertion : No typesetting found for |- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		recsex	Could not format ( ( C e. No /\ C =/= 0s ) -> E. x e. No ( C x.s x ) = 1s ) : No typesetting found for |- ( ( C e. No /\ C =/= 0s ) -> E. x e. No ( C x.s x ) = 1s ) with typecode |-
2	1	3ad2ant3	Could not format ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> E. x e. No ( C x.s x ) = 1s ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> E. x e. No ( C x.s x ) = 1s ) with typecode |-
3		divsmulw	Could not format ( ( ( 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 ) ) : No typesetting found for |- ( ( ( 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 ) ) with typecode |-
4	2 3	mpdan	Could not format ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) : No typesetting found for |- ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) with typecode |-