Theorem ringacl

Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014)

	Ref	Expression
Hypotheses	ringacl.b	$$⊢B={\mathrm{Base}}_R$$
	ringacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
Assertion	ringacl	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		ringacl.b	$$⊢B={\mathrm{Base}}_R$$
2		ringacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
3		ringgrp	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Grp}$$
4	1 2	grpcl	$$⊢\left(R\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$
5	3 4	syl3an1	$$⊢\left(R\in \mathrm{Ring}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$