Theorem grpinvcld

Description: A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025)

	Ref	Expression
Hypotheses	grpinvcld.b	$$⊢B={\mathrm{Base}}_G$$
	grpinvcld.n	$$⊢N={inv}_g\left(G\right)$$
	grpinvcld.g	$$⊢\phi \to G\in \mathrm{Grp}$$
	grpinvcld.1	$$⊢\phi \to X\in B$$
Assertion	grpinvcld	$$⊢\phi \to N\left(X\right)\in B$$

Proof

Step	Hyp	Ref	Expression
1		grpinvcld.b	$$⊢B={\mathrm{Base}}_G$$
2		grpinvcld.n	$$⊢N={inv}_g\left(G\right)$$
3		grpinvcld.g	$$⊢\phi \to G\in \mathrm{Grp}$$
4		grpinvcld.1	$$⊢\phi \to X\in B$$
5	1 2	grpinvcl	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\right)\to N\left(X\right)\in B$$
6	3 4 5	syl2anc	$$⊢\phi \to N\left(X\right)\in B$$