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 )