Theorem sltsubaddd

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

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

Proof

Step	Hyp	Ref	Expression
1		sltsubadd.1	|- ( ph -> A e. No )
2		sltsubadd.2	|- ( ph -> B e. No )
3		sltsubadd.3	|- ( ph -> C e. No )
4	1 2	subscld	|- ( ph -> ( A -s B ) e. No )
5	4 3 2	sltadd1d	|- ( ph -> ( ( A -s B ) ( ( A -s B ) +s B )
6		npcans	|- ( ( 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