Theorem ringcl

Description: Closure of the multiplication operation of a ring. (Contributed by NM, 26-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	ringcl	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{·}Y\in B$$

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	mndcl	$$⊢\left({\mathrm{mulGrp}}_R\in \mathrm{Mnd}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{·}Y\in B$$
8	4 7	syl3an1	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{·}Y\in B$$