Theorem srglidm

Description: The unit element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	srgidm.b	$$⊢B={\mathrm{Base}}_R$$
	srgidm.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	srgidm.u	$$⊢\underset{\dot{}}{1}=1_R$$
Assertion	srglidm	$$⊢\left(R\in \mathrm{SRing}\wedge X\in B\right)\to \underset{\dot{}}{1}\underset{\dot{}}{·}X=X$$

Proof

Step	Hyp	Ref	Expression
1		srgidm.b	$$⊢B={\mathrm{Base}}_R$$
2		srgidm.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		srgidm.u	$$⊢\underset{\dot{}}{1}=1_R$$
4	1 2 3	srgidmlem	$$⊢\left(R\in \mathrm{SRing}\wedge X\in B\right)\to \left(\underset{\dot{}}{1}\underset{\dot{}}{·}X=X\wedge X\underset{\dot{}}{·}\underset{\dot{}}{1}=X\right)$$
5	4	simpld	$$⊢\left(R\in \mathrm{SRing}\wedge X\in B\right)\to \underset{\dot{}}{1}\underset{\dot{}}{·}X=X$$