Theorem sltneg

Description: Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	sltneg	|- ( ( A e. No /\ B e. No ) -> ( A ( -us ` B )

Proof

Step	Hyp	Ref	Expression
1		sltnegim	|- ( ( A e. No /\ B e. No ) -> ( A ( -us ` B )
2		negscl	|- ( B e. No -> ( -us ` B ) e. No )
3		negscl	|- ( A e. No -> ( -us ` A ) e. No )
4		sltnegim	|- ( ( ( -us ` B ) e. No /\ ( -us ` A ) e. No ) -> ( ( -us ` B ) ( -us ` ( -us ` A ) )
5	2 3 4	syl2anr	|- ( ( A e. No /\ B e. No ) -> ( ( -us ` B ) ( -us ` ( -us ` A ) )
6		negnegs	|- ( A e. No -> ( -us ` ( -us ` A ) ) = A )
7		negnegs	|- ( B e. No -> ( -us ` ( -us ` B ) ) = B )
8	6 7	breqan12d	|- ( ( A e. No /\ B e. No ) -> ( ( -us ` ( -us ` A ) ) A
9	5 8	sylibd	|- ( ( A e. No /\ B e. No ) -> ( ( -us ` B ) A
10	1 9	impbid	|- ( ( A e. No /\ B e. No ) -> ( A ( -us ` B )