Theorem grpcl

Description: Closure of the operation of a group. (Contributed by NM, 14-Aug-2011)

	Ref	Expression
Hypotheses	grpcl.b	$$⊢B={\mathrm{Base}}_G$$
	grpcl.p	$$⊢\underset{\dot{}}{+}={+}_G$$
Assertion	grpcl	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	$$⊢B={\mathrm{Base}}_G$$
2		grpcl.p	$$⊢\underset{\dot{}}{+}={+}_G$$
3		grpmnd	$$⊢G\in \mathrm{Grp}\to G\in \mathrm{Mnd}$$
4	1 2	mndcl	$$⊢\left(G\in \mathrm{Mnd}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$
5	3 4	syl3an1	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$