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 ∈ No → ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )
2	1	negsidd	⊢ ( 𝐴 ∈ No → ( ( -us ‘ 𝐴 ) +s ( -us ‘ ( -us ‘ 𝐴 ) ) ) = 0s )
3	1	negscld	⊢ ( 𝐴 ∈ No → ( -us ‘ ( -us ‘ 𝐴 ) ) ∈ No )
4	3 1	addscomd	⊢ ( 𝐴 ∈ No → ( ( -us ‘ ( -us ‘ 𝐴 ) ) +s ( -us ‘ 𝐴 ) ) = ( ( -us ‘ 𝐴 ) +s ( -us ‘ ( -us ‘ 𝐴 ) ) ) )
5		negsid	⊢ ( 𝐴 ∈ No → ( 𝐴 +s ( -us ‘ 𝐴 ) ) = 0s )
6	2 4 5	3eqtr4d	⊢ ( 𝐴 ∈ No → ( ( -us ‘ ( -us ‘ 𝐴 ) ) +s ( -us ‘ 𝐴 ) ) = ( 𝐴 +s ( -us ‘ 𝐴 ) ) )
7		id	⊢ ( 𝐴 ∈ No → 𝐴 ∈ No )
8	3 7 1	addscan2d	⊢ ( 𝐴 ∈ No → ( ( ( -us ‘ ( -us ‘ 𝐴 ) ) +s ( -us ‘ 𝐴 ) ) = ( 𝐴 +s ( -us ‘ 𝐴 ) ) ↔ ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 ) )
9	6 8	mpbid	⊢ ( 𝐴 ∈ No → ( -us ‘ ( -us ‘ 𝐴 ) ) = 𝐴 )