Theorem ltsubadds2d

Description: Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025)

	Ref	Expression
Hypotheses	ltsubadds.1	|- ( ph -> A e. No )
	ltsubadds.2	|- ( ph -> B e. No )
	ltsubadds.3	|- ( ph -> C e. No )
Assertion	ltsubadds2d	|- ( ph -> ( ( A -s B ) A

Proof

Step	Hyp	Ref	Expression
1		ltsubadds.1	|- ( ph -> A e. No )
2		ltsubadds.2	|- ( ph -> B e. No )
3		ltsubadds.3	|- ( ph -> C e. No )
4	1 2 3	ltsubaddsd	|- ( ph -> ( ( A -s B ) A
5	2 3	addscomd	|- ( ph -> ( B +s C ) = ( C +s B ) )
6	5	breq2d	|- ( ph -> ( A A
7	4 6	bitr4d	|- ( ph -> ( ( A -s B ) A