Theorem negnegs

Description: A surreal is equal to the negative of its negative. Theorem 4(ii) of Conway p. 17. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negnegs	|- ( A e. No -> ( -us ` ( -us ` A ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		negscl	|- ( A e. No -> ( -us ` A ) e. No )
2	1	negsidd	|- ( A e. No -> ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) = 0s )
3	1	negscld	|- ( A e. No -> ( -us ` ( -us ` A ) ) e. No )
4	3 1	addscomd	|- ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) )
5		negsid	|- ( A e. No -> ( A +s ( -us ` A ) ) = 0s )
6	2 4 5	3eqtr4d	|- ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) )
7		id	|- ( A e. No -> A e. No )
8	3 7 1	addscan2d	|- ( A e. No -> ( ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) <-> ( -us ` ( -us ` A ) ) = A ) )
9	6 8	mpbid	|- ( A e. No -> ( -us ` ( -us ` A ) ) = A )