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Metamath Proof Explorer

Theorem grpidcld

Description: The identity element of a group belongs to the group. (Contributed by Thierry Arnoux, 4-May-2026)

Ref Expression
Hypotheses grpidcld.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpidcld.2 ⊢ 0 = ( 0g ‘ 𝐺 )
grpidcld.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
Assertion grpidcld ( 𝜑 → 0 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 grpidcld.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpidcld.2 ⊢ 0 = ( 0g ‘ 𝐺 )
3 grpidcld.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 1 2 grpidcl ⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )
5 3 4 syl ⊢ ( 𝜑 → 0 ∈ 𝐵 )