Theorem sleadd2

Description: Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Assertion	sleadd2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( C +s A ) <_s ( C +s B ) ) )

Proof

Step	Hyp	Ref	Expression
1		sleadd1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) )
2		addscom	|- ( ( A e. No /\ C e. No ) -> ( A +s C ) = ( C +s A ) )
3	2	3adant2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A +s C ) = ( C +s A ) )
4		addscom	|- ( ( B e. No /\ C e. No ) -> ( B +s C ) = ( C +s B ) )
5	4	3adant1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( B +s C ) = ( C +s B ) )
6	3 5	breq12d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A +s C ) <_s ( B +s C ) <-> ( C +s A ) <_s ( C +s B ) ) )
7	1 6	bitrd	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( C +s A ) <_s ( C +s B ) ) )