Theorem muls02

Description: Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025)

	Ref	Expression
Assertion	muls02	⊢ ( 𝐴 ∈ No → ( 0s ·s 𝐴 ) = 0s )

Proof

Step	Hyp	Ref	Expression
1		0no	⊢ 0s ∈ No
2		mulscom	⊢ ( ( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ·s 𝐴 ) = ( 𝐴 ·s 0s ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ No → ( 0s ·s 𝐴 ) = ( 𝐴 ·s 0s ) )
4		muls01	⊢ ( 𝐴 ∈ No → ( 𝐴 ·s 0s ) = 0s )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ No → ( 0s ·s 𝐴 ) = 0s )