Theorem grpinvcld

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

	Ref	Expression
Hypotheses	grpinvcld.b	|- B = ( Base ` G )
	grpinvcld.n	|- N = ( invg ` G )
	grpinvcld.g	|- ( ph -> G e. Grp )
	grpinvcld.1	|- ( ph -> X e. B )
Assertion	grpinvcld	|- ( ph -> ( N ` X ) e. B )

Proof

Step	Hyp	Ref	Expression
1		grpinvcld.b	|- B = ( Base ` G )
2		grpinvcld.n	|- N = ( invg ` G )
3		grpinvcld.g	|- ( ph -> G e. Grp )
4		grpinvcld.1	|- ( ph -> X e. B )
5	1 2	grpinvcl	|- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
6	3 4 5	syl2anc	|- ( ph -> ( N ` X ) e. B )