Theorem sltsub1

Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	sltsub1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( A -s C )

Proof

Step	Hyp	Ref	Expression
1		negscl	|- ( C e. No -> ( -us ` C ) e. No )
2		sltadd1	|- ( ( A e. No /\ B e. No /\ ( -us ` C ) e. No ) -> ( A ( A +s ( -us ` C ) )
3	1 2	syl3an3	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( A +s ( -us ` C ) )
4		subsval	|- ( ( A e. No /\ C e. No ) -> ( A -s C ) = ( A +s ( -us ` C ) ) )
5	4	3adant2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A -s C ) = ( A +s ( -us ` C ) ) )
6		subsval	|- ( ( B e. No /\ C e. No ) -> ( B -s C ) = ( B +s ( -us ` C ) ) )
7	6	3adant1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( B -s C ) = ( B +s ( -us ` C ) ) )
8	5 7	breq12d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A -s C ) ( A +s ( -us ` C ) )
9	3 8	bitr4d	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A ( A -s C )