Theorem grpinvcl

Description: A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011) (Revised by Mario Carneiro, 4-May-2015)

	Ref	Expression
Hypotheses	grpinvcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpinvcl.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpinvcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpinvcl.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
3	1 2	grpinvf	⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 )
4	3	ffvelrnda	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )