Theorem addslid

Description: Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	addslid	⊢ ( 𝐴 ∈ No → ( 0s +s 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 ∈ No → 𝐴 ∈ No )
2		0sno	⊢ 0s ∈ No
3	2	a1i	⊢ ( 𝐴 ∈ No → 0s ∈ No )
4	1 3	addscomd	⊢ ( 𝐴 ∈ No → ( 𝐴 +s 0s ) = ( 0s +s 𝐴 ) )
5		addsrid	⊢ ( 𝐴 ∈ No → ( 𝐴 +s 0s ) = 𝐴 )
6	4 5	eqtr3d	⊢ ( 𝐴 ∈ No → ( 0s +s 𝐴 ) = 𝐴 )