Theorem sleadd2im

Description: Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	sleadd2im	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) <_s ( C +s B ) -> A <_s 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	breq12d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) <-> ( C +s A ) <_s ( C +s B ) ) )
6		sleadd1im	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) -> A <_s B ) )
7	5 6	sylbird	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( C +s A ) <_s ( C +s B ) -> A <_s B ) )