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Metamath Proof Explorer

Theorem ablogrpo

Description: An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006) (New usage is discouraged.)

Ref Expression
Assertion ablogrpo ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ran 𝐺 = ran 𝐺
2 1 isablo ⊢ ( 𝐺 ∈ AbelOp ↔ ( 𝐺 ∈ GrpOp ∧ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( 𝑥 𝐺 𝑦 ) = ( 𝑦 𝐺 𝑥 ) ) )
3 2 simplbi ⊢ ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp )