Theorem isabli

Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011)

	Ref	Expression
Hypotheses	isabli.g	⊢ 𝐺 ∈ Grp
	isabli.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	isabli.p	⊢ + = ( +g ‘ 𝐺 )
	isabli.c	⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
Assertion	isabli	⊢ 𝐺 ∈ Abel

Proof

Step	Hyp	Ref	Expression
1		isabli.g	⊢ 𝐺 ∈ Grp
2		isabli.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		isabli.p	⊢ + = ( +g ‘ 𝐺 )
4		isabli.c	⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
5	4	rgen2	⊢ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 )
6	2 3	isabl2	⊢ ( 𝐺 ∈ Abel ↔ ( 𝐺 ∈ Grp ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
7	1 5 6	mpbir2an	⊢ 𝐺 ∈ Abel