Theorem isringid

Description: Properties showing that an element I is the unity element of a ring. (Contributed by NM, 7-Aug-2013)

	Ref	Expression
Hypotheses	ringidm.b	$$⊢B={\mathrm{Base}}_R$$
	ringidm.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringidm.u	$$⊢\underset{\dot{}}{1}=1_R$$
Assertion	isringid	$$⊢R\in \mathrm{Ring}\to \left(\left(I\in B\wedge \forall x\in B\left(I\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}I=x\right)\right)\leftrightarrow \underset{\dot{}}{1}=I\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringidm.b	$$⊢B={\mathrm{Base}}_R$$
2		ringidm.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringidm.u	$$⊢\underset{\dot{}}{1}=1_R$$
4		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
5	4 1	mgpbas	$$⊢B={\mathrm{Base}}_{{\mathrm{mulGrp}}_R}$$
6	4 3	ringidval	$$⊢\underset{\dot{}}{1}=0_{{\mathrm{mulGrp}}_R}$$
7	4 2	mgpplusg	$$⊢\underset{\dot{}}{·}={+}_{{\mathrm{mulGrp}}_R}$$
8	1 2	ringideu	$$⊢R\in \mathrm{Ring}\to \exists !y\in B\forall x\in B\left(y\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}y=x\right)$$
9		reurex	$$⊢\exists !y\in B\forall x\in B\left(y\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}y=x\right)\to \exists y\in B\forall x\in B\left(y\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}y=x\right)$$
10	8 9	syl	$$⊢R\in \mathrm{Ring}\to \exists y\in B\forall x\in B\left(y\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}y=x\right)$$
11	5 6 7 10	ismgmid	$$⊢R\in \mathrm{Ring}\to \left(\left(I\in B\wedge \forall x\in B\left(I\underset{\dot{}}{·}x=x\wedge x\underset{\dot{}}{·}I=x\right)\right)\leftrightarrow \underset{\dot{}}{1}=I\right)$$