Theorem ablcom

Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011)

	Ref	Expression
Hypotheses	ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablcom.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablcom.p	⊢ + = ( +g ‘ 𝐺 )
3		ablcmn	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ CMnd )
4	1 2	cmncom	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
5	3 4	syl3an1	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )