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 = ( Base ‘ 𝐺 )
	grpidcl.o	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grpidcl	⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpidcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpidcl.o	⊢ 0 = ( 0g ‘ 𝐺 )
3		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
4	1 2	mndidcl	⊢ ( 𝐺 ∈ Mnd → 0 ∈ 𝐵 )
5	3 4	syl	⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )