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	$$⊢B={\mathrm{Base}}_G$$
	grpidcld.2	$$⊢\underset{\dot{}}{0}=0_G$$
	grpidcld.3	$$⊢\phi \to G\in \mathrm{Grp}$$
Assertion	grpidcld	$$⊢\phi \to \underset{\dot{}}{0}\in B$$

Proof

Step	Hyp	Ref	Expression
1		grpidcld.1	$$⊢B={\mathrm{Base}}_G$$
2		grpidcld.2	$$⊢\underset{\dot{}}{0}=0_G$$
3		grpidcld.3	$$⊢\phi \to G\in \mathrm{Grp}$$
4	1 2	grpidcl	$$⊢G\in \mathrm{Grp}\to \underset{\dot{}}{0}\in B$$
5	3 4	syl	$$⊢\phi \to \underset{\dot{}}{0}\in B$$