Theorem srgideu

Description: The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	srgcl.b	$$⊢B={\mathrm{Base}}_R$$
	srgcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	srgideu	$$⊢R\in \mathrm{SRing}\to \exists !u\in B\forall x\in B\left(u\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}u=x\right)$$

Proof

Step	Hyp	Ref	Expression
1		srgcl.b	$$⊢B={\mathrm{Base}}_R$$
2		srgcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
4	3	srgmgp	$$⊢R\in \mathrm{SRing}\to {\mathrm{mulGrp}}_R\in \mathrm{Mnd}$$
5	3 1	mgpbas	$$⊢B={\mathrm{Base}}_{{\mathrm{mulGrp}}_R}$$
6	3 2	mgpplusg	$$⊢\underset{\dot{}}{·}={+}_{{\mathrm{mulGrp}}_R}$$
7	5 6	mndideu	$$⊢{\mathrm{mulGrp}}_R\in \mathrm{Mnd}\to \exists !u\in B\forall x\in B\left(u\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}u=x\right)$$
8	4 7	syl	$$⊢R\in \mathrm{SRing}\to \exists !u\in B\forall x\in B\left(u\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}u=x\right)$$