Theorem grpidcl

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

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

Proof

Step	Hyp	Ref	Expression
1		grpidcl.b	$$⊢B={\mathrm{Base}}_G$$
2		grpidcl.o	$$⊢\underset{\dot{}}{0}=0_G$$
3		grpmnd	$$⊢G\in \mathrm{Grp}\to G\in \mathrm{Mnd}$$
4	1 2	mndidcl	$$⊢G\in \mathrm{Mnd}\to \underset{\dot{}}{0}\in B$$
5	3 4	syl	$$⊢G\in \mathrm{Grp}\to \underset{\dot{}}{0}\in B$$