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 )