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Metamath Proof Explorer

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 )