Theorem mulslid

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

	Ref	Expression
Assertion	mulslid	$$⊢A\in \mathrm{No}\to 1_s{\cdot }_{s}A=A$$

Proof

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