Theorem ablcom

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

	Ref	Expression
Hypotheses	ablcom.b	$$⊢B={\mathrm{Base}}_G$$
	ablcom.p	$$⊢\underset{\dot{}}{+}={+}_G$$
Assertion	ablcom	$$⊢\left(G\in \mathrm{Abel}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y=Y\underset{\dot{}}{+}X$$

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	$$⊢B={\mathrm{Base}}_G$$
2		ablcom.p	$$⊢\underset{\dot{}}{+}={+}_G$$
3		ablcmn	$$⊢G\in \mathrm{Abel}\to G\in \mathrm{CMnd}$$
4	1 2	cmncom	$$⊢\left(G\in \mathrm{CMnd}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y=Y\underset{\dot{}}{+}X$$
5	3 4	syl3an1	$$⊢\left(G\in \mathrm{Abel}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y=Y\underset{\dot{}}{+}X$$