Theorem ringideu

Description: The unity element of a ring is unique. (Contributed by NM, 4-Apr-2009) (Revised by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	$$⊢B={\mathrm{Base}}_R$$
2		ringcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
4	3	ringmgp	$$⊢R\in \mathrm{Ring}\to {\mathrm{mulGrp}}_R\in \mathrm{Mnd}$$
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	$$⊢R\in \mathrm{Ring}\to \exists !u\in B\forall x\in B\left(u\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}u=x\right)$$