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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 = ( 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 ffvelcdmda ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )