Theorem posdifsd

Description: Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	posdifsd.1	|- ( ph -> A e. No )
	posdifsd.2	|- ( ph -> B e. No )
Assertion	posdifsd	|- ( ph -> ( A 0s

Proof

Step	Hyp	Ref	Expression
1		posdifsd.1	|- ( ph -> A e. No )
2		posdifsd.2	|- ( ph -> B e. No )
3		0sno	|- 0s e. No
4	3	a1i	|- ( ph -> 0s e. No )
5	2 1	subscld	|- ( ph -> ( B -s A ) e. No )
6	4 5 1	sltadd1d	|- ( ph -> ( 0s ( 0s +s A )
7		addslid	|- ( A e. No -> ( 0s +s A ) = A )
8	1 7	syl	|- ( ph -> ( 0s +s A ) = A )
9		npcans	|- ( ( B e. No /\ A e. No ) -> ( ( B -s A ) +s A ) = B )
10	2 1 9	syl2anc	|- ( ph -> ( ( B -s A ) +s A ) = B )
11	8 10	breq12d	|- ( ph -> ( ( 0s +s A ) A
12	6 11	bitr2d	|- ( ph -> ( A 0s