Theorem ltaddsubsd

Description: Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025)

	Ref	Expression
Hypotheses	ltsubadds.1	|- ( ph -> A e. No )
	ltsubadds.2	|- ( ph -> B e. No )
	ltsubadds.3	|- ( ph -> C e. No )
Assertion	ltaddsubsd	|- ( 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	addscld	|- ( ph -> ( A +s B ) e. No )
5	4 3 2	ltsubs1d	|- ( ph -> ( ( A +s B ) ( ( A +s B ) -s B )
6		pncans	|- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) -s B ) = A )
7	1 2 6	syl2anc	|- ( ph -> ( ( A +s B ) -s B ) = A )
8	7	breq1d	|- ( ph -> ( ( ( A +s B ) -s B ) A
9	5 8	bitrd	|- ( ph -> ( ( A +s B ) A