Theorem addscan1

Description: Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	addscan1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) = ( C +s B ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		addscom	|- ( ( A e. No /\ C e. No ) -> ( A +s C ) = ( C +s A ) )
2	1	3adant2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A +s C ) = ( C +s A ) )
3		addscom	|- ( ( B e. No /\ C e. No ) -> ( B +s C ) = ( C +s B ) )
4	3	3adant1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( B +s C ) = ( C +s B ) )
5	2 4	eqeq12d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) = ( B +s C ) <-> ( C +s A ) = ( C +s B ) ) )
6		addscan2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) = ( B +s C ) <-> A = B ) )
7	5 6	bitr3d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) = ( C +s B ) <-> A = B ) )