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	|- ( ph -> G e. Grp )
Assertion	grpidcld	|- ( ph -> .0. e. B )

Proof

Step	Hyp	Ref	Expression
1		grpidcld.1	|- B = ( Base ` G )
2		grpidcld.2	|- .0. = ( 0g ` G )
3		grpidcld.3	|- ( ph -> G e. Grp )
4	1 2	grpidcl	|- ( G e. Grp -> .0. e. B )
5	3 4	syl	|- ( ph -> .0. e. B )