Theorem negsidd

Description: Surreal addition of a number and its negative. Theorem 4(iii) of Conway p. 17. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypothesis	negsidd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
Assertion	negsidd	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ 𝐴 ) ) = 0s )

Proof

Step	Hyp	Ref	Expression
1		negsidd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		negsid	⊢ ( 𝐴 ∈ No → ( 𝐴 +s ( -us ‘ 𝐴 ) ) = 0s )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ 𝐴 ) ) = 0s )