Theorem muls02

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

	Ref	Expression
Assertion	muls02	$$⊢A\in \mathrm{No}\to 0_s{\cdot }_{s}A=0_s$$

Proof

Step	Hyp	Ref	Expression
1		0no	$$⊢0_s\in \mathrm{No}$$
2		mulscom	$$⊢\left(0_s\in \mathrm{No}\wedge A\in \mathrm{No}\right)\to 0_s{\cdot }_{s}A=A{\cdot }_s{0}_s$$
3	1 2	mpan	$$⊢A\in \mathrm{No}\to 0_s{\cdot }_{s}A=A{\cdot }_s{0}_s$$
4		muls01	$$⊢A\in \mathrm{No}\to A{\cdot }_s{0}_s=0_s$$
5	3 4	eqtrd	$$⊢A\in \mathrm{No}\to 0_s{\cdot }_{s}A=0_s$$