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Metamath Proof Explorer

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 = inv g ⁡ 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 = inv g ⁡ G
3 1 2 grpinvf ⊢ G ∈ Grp → N : B ⟶ B
4 3 ffvelcdmda ⊢ G ∈ Grp ∧ X ∈ B → N ⁡ X ∈ B