Theorem no2times

Description: Version of 2times for surreal numbers. (Contributed by Scott Fenton, 23-Jul-2025)

	Ref	Expression
Assertion	no2times	⊢ ( 𝐴 ∈ No → ( 2s ·s 𝐴 ) = ( 𝐴 +s 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		1p1e2s	⊢ ( 1s +s 1s ) = 2s
2	1	oveq1i	⊢ ( ( 1s +s 1s ) ·s 𝐴 ) = ( 2s ·s 𝐴 )
3	2	eqcomi	⊢ ( 2s ·s 𝐴 ) = ( ( 1s +s 1s ) ·s 𝐴 )
4		1sno	⊢ 1s ∈ No
5	4	a1i	⊢ ( 𝐴 ∈ No → 1s ∈ No )
6		id	⊢ ( 𝐴 ∈ No → 𝐴 ∈ No )
7	5 5 6	addsdird	⊢ ( 𝐴 ∈ No → ( ( 1s +s 1s ) ·s 𝐴 ) = ( ( 1s ·s 𝐴 ) +s ( 1s ·s 𝐴 ) ) )
8		mulslid	⊢ ( 𝐴 ∈ No → ( 1s ·s 𝐴 ) = 𝐴 )
9	8 8	oveq12d	⊢ ( 𝐴 ∈ No → ( ( 1s ·s 𝐴 ) +s ( 1s ·s 𝐴 ) ) = ( 𝐴 +s 𝐴 ) )
10	7 9	eqtrd	⊢ ( 𝐴 ∈ No → ( ( 1s +s 1s ) ·s 𝐴 ) = ( 𝐴 +s 𝐴 ) )
11	3 10	eqtrid	⊢ ( 𝐴 ∈ No → ( 2s ·s 𝐴 ) = ( 𝐴 +s 𝐴 ) )