Theorem mndcl

Description: Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Proof shortened by AV, 8-Feb-2020)

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

Proof

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