Theorem mndidcl

Description: The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 27-Dec-2014)

	Ref	Expression
Hypotheses	mndidcl.b	$$⊢B={\mathrm{Base}}_G$$
	mndidcl.o	$$⊢\underset{\dot{}}{0}=0_G$$
Assertion	mndidcl	$$⊢G\in \mathrm{Mnd}\to \underset{\dot{}}{0}\in B$$

Proof

Step	Hyp	Ref	Expression
1		mndidcl.b	$$⊢B={\mathrm{Base}}_G$$
2		mndidcl.o	$$⊢\underset{\dot{}}{0}=0_G$$
3		eqid	$$⊢{+}_G={+}_G$$
4	1 3	mndid	$$⊢G\in \mathrm{Mnd}\to \exists x\in B\forall y\in B\left(x{+}_{G}y=y\wedge y{+}_{G}x=y\right)$$
5	1 2 3 4	mgmidcl	$$⊢G\in \mathrm{Mnd}\to \underset{\dot{}}{0}\in B$$