Theorem mulslid

Description: Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	mulslid	⊢ ( 𝐴 ∈ No → ( 1s ·s 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		1sno	⊢ 1s ∈ No
2		mulscom	⊢ ( ( 1s ∈ No ∧ 𝐴 ∈ No ) → ( 1s ·s 𝐴 ) = ( 𝐴 ·s 1s ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ No → ( 1s ·s 𝐴 ) = ( 𝐴 ·s 1s ) )
4		mulsrid	⊢ ( 𝐴 ∈ No → ( 𝐴 ·s 1s ) = 𝐴 )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ No → ( 1s ·s 𝐴 ) = 𝐴 )