Theorem sleadd1d

Description: Addition to both sides of surreal less-than or equal. Theorem 5 of Conway p. 18. (Contributed by Scott Fenton, 21-Jan-2025)

	Ref	Expression
Hypotheses	addscand.1	|- ( ph -> A e. No )
	addscand.2	|- ( ph -> B e. No )
	addscand.3	|- ( ph -> C e. No )
Assertion	sleadd1d	|- ( ph -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addscand.1	|- ( ph -> A e. No )
2		addscand.2	|- ( ph -> B e. No )
3		addscand.3	|- ( ph -> C e. No )
4		sleadd1	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A <_s B <-> ( A +s C ) <_s ( B +s C ) ) )