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	Could not format assertion : No typesetting found for |- ( A e. No -> ( -us ` ( -us ` A ) ) = A ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		negscl	Could not format ( A e. No -> ( -us ` A ) e. No ) : No typesetting found for |- ( A e. No -> ( -us ` A ) e. No ) with typecode |-
2	1	negsidd	Could not format ( A e. No -> ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) = 0s ) : No typesetting found for |- ( A e. No -> ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) = 0s ) with typecode |-
3	1	negscld	Could not format ( A e. No -> ( -us ` ( -us ` A ) ) e. No ) : No typesetting found for |- ( A e. No -> ( -us ` ( -us ` A ) ) e. No ) with typecode |-
4	3 1	addscomd	Could not format ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) ) : No typesetting found for |- ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( ( -us ` A ) +s ( -us ` ( -us ` A ) ) ) ) with typecode |-
5		negsid	Could not format ( A e. No -> ( A +s ( -us ` A ) ) = 0s ) : No typesetting found for |- ( A e. No -> ( A +s ( -us ` A ) ) = 0s ) with typecode |-
6	2 4 5	3eqtr4d	Could not format ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) ) : No typesetting found for |- ( A e. No -> ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) ) with typecode |-
7		id	$$⊢A\in \mathrm{No}\to A\in \mathrm{No}$$
8	3 7 1	addscan2d	Could not format ( A e. No -> ( ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) <-> ( -us ` ( -us ` A ) ) = A ) ) : No typesetting found for |- ( A e. No -> ( ( ( -us ` ( -us ` A ) ) +s ( -us ` A ) ) = ( A +s ( -us ` A ) ) <-> ( -us ` ( -us ` A ) ) = A ) ) with typecode |-
9	6 8	mpbid	Could not format ( A e. No -> ( -us ` ( -us ` A ) ) = A ) : No typesetting found for |- ( A e. No -> ( -us ` ( -us ` A ) ) = A ) with typecode |-