Theorem sltsubsub2bd

Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025)

	Ref	Expression
Hypotheses	sltsubsubbd.1	|- ( ph -> A e. No )
	sltsubsubbd.2	|- ( ph -> B e. No )
	sltsubsubbd.3	|- ( ph -> C e. No )
	sltsubsubbd.4	|- ( ph -> D e. No )
Assertion	sltsubsub2bd	|- ( ph -> ( ( A -s B ) ( D -s C )

Proof

Step	Hyp	Ref	Expression
1		sltsubsubbd.1	|- ( ph -> A e. No )
2		sltsubsubbd.2	|- ( ph -> B e. No )
3		sltsubsubbd.3	|- ( ph -> C e. No )
4		sltsubsubbd.4	|- ( ph -> D e. No )
5	4 3	subscld	|- ( ph -> ( D -s C ) e. No )
6	2 1	subscld	|- ( ph -> ( B -s A ) e. No )
7	5 6	sltnegd	|- ( ph -> ( ( D -s C ) ( -us ` ( B -s A ) )
8	2 1	negsubsdi2d	|- ( ph -> ( -us ` ( B -s A ) ) = ( A -s B ) )
9	4 3	negsubsdi2d	|- ( ph -> ( -us ` ( D -s C ) ) = ( C -s D ) )
10	8 9	breq12d	|- ( ph -> ( ( -us ` ( B -s A ) ) ( A -s B )
11	7 10	bitr2d	|- ( ph -> ( ( A -s B ) ( D -s C )