Theorem sleadd2d

Description: Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	addscand.1	|- ( ph -> A e. No )
	addscand.2	|- ( ph -> B e. No )
	addscand.3	|- ( ph -> C e. No )
Assertion	sleadd2d	|- ( ph -> ( A <_s B <-> ( C +s A ) <_s ( C +s B ) ) )

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		sleadd2	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( A <_s B <-> ( C +s A ) <_s ( C +s B ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A <_s B <-> ( C +s A ) <_s ( C +s B ) ) )