Theorem negsval2

Description: Surreal negation in terms of subtraction. (Contributed by Scott Fenton, 15-Apr-2025)

	Ref	Expression
Assertion	negsval2	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) = ( 0s -s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0sno	⊢ 0s ∈ No
2		subsval	⊢ ( ( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s -s 𝐴 ) = ( 0s +s ( -us ‘ 𝐴 ) ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ No → ( 0s -s 𝐴 ) = ( 0s +s ( -us ‘ 𝐴 ) ) )
4		negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )
5		addslid	⊢ ( ( -us ‘ 𝐴 ) ∈ No → ( 0s +s ( -us ‘ 𝐴 ) ) = ( -us ‘ 𝐴 ) )
6	4 5	syl	⊢ ( 𝐴 ∈ No → ( 0s +s ( -us ‘ 𝐴 ) ) = ( -us ‘ 𝐴 ) )
7	3 6	eqtr2d	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) = ( 0s -s 𝐴 ) )