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| 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 | 
|---|---|
| grpinvcl.b | ⊢ 𝐵 = (Base‘𝐺) | 
| grpinvcl.n | ⊢ 𝑁 = (invg‘𝐺) | 
| Ref | Expression | 
|---|---|
| grpinvcl | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grpinvcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | grpinvcl.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
| 3 | 1, 2 | grpinvf 19005 | . 2 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵) | 
| 4 | 3 | ffvelcdmda 7103 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) | 
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