Theorem negsdi

Description: Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Assertion	negsdi	|- ( ( A e. No /\ B e. No ) -> ( -us ` ( A +s B ) ) = ( ( -us ` A ) +s ( -us ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		addscl	|- ( ( A e. No /\ B e. No ) -> ( A +s B ) e. No )
2	1	negsidd	|- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) +s ( -us ` ( A +s B ) ) ) = 0s )
3		negsid	|- ( A e. No -> ( A +s ( -us ` A ) ) = 0s )
4		negsid	|- ( B e. No -> ( B +s ( -us ` B ) ) = 0s )
5	3 4	oveqan12d	|- ( ( A e. No /\ B e. No ) -> ( ( A +s ( -us ` A ) ) +s ( B +s ( -us ` B ) ) ) = ( 0s +s 0s ) )
6		0sno	|- 0s e. No
7		addslid	|- ( 0s e. No -> ( 0s +s 0s ) = 0s )
8	6 7	ax-mp	|- ( 0s +s 0s ) = 0s
9	5 8	eqtr2di	|- ( ( A e. No /\ B e. No ) -> 0s = ( ( A +s ( -us ` A ) ) +s ( B +s ( -us ` B ) ) ) )
10		simpl	|- ( ( A e. No /\ B e. No ) -> A e. No )
11	10	negscld	|- ( ( A e. No /\ B e. No ) -> ( -us ` A ) e. No )
12		simpr	|- ( ( A e. No /\ B e. No ) -> B e. No )
13	12	negscld	|- ( ( A e. No /\ B e. No ) -> ( -us ` B ) e. No )
14	10 11 12 13	adds4d	|- ( ( A e. No /\ B e. No ) -> ( ( A +s ( -us ` A ) ) +s ( B +s ( -us ` B ) ) ) = ( ( A +s B ) +s ( ( -us ` A ) +s ( -us ` B ) ) ) )
15	2 9 14	3eqtrd	|- ( ( A e. No /\ B e. No ) -> ( ( A +s B ) +s ( -us ` ( A +s B ) ) ) = ( ( A +s B ) +s ( ( -us ` A ) +s ( -us ` B ) ) ) )
16	1	negscld	|- ( ( A e. No /\ B e. No ) -> ( -us ` ( A +s B ) ) e. No )
17		negscl	|- ( A e. No -> ( -us ` A ) e. No )
18		negscl	|- ( B e. No -> ( -us ` B ) e. No )
19		addscl	|- ( ( ( -us ` A ) e. No /\ ( -us ` B ) e. No ) -> ( ( -us ` A ) +s ( -us ` B ) ) e. No )
20	17 18 19	syl2an	|- ( ( A e. No /\ B e. No ) -> ( ( -us ` A ) +s ( -us ` B ) ) e. No )
21	16 20 1	addscan1d	|- ( ( A e. No /\ B e. No ) -> ( ( ( A +s B ) +s ( -us ` ( A +s B ) ) ) = ( ( A +s B ) +s ( ( -us ` A ) +s ( -us ` B ) ) ) <-> ( -us ` ( A +s B ) ) = ( ( -us ` A ) +s ( -us ` B ) ) ) )
22	15 21	mpbid	|- ( ( A e. No /\ B e. No ) -> ( -us ` ( A +s B ) ) = ( ( -us ` A ) +s ( -us ` B ) ) )