crngcom

Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	$⊢ B = {Base}_{R}$
	ringcl.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
Assertion	crngcom	$⊢ (R \in CRing \land X \in B \land Y \in B) \to X \underset{˙}{\cdot} Y = Y \underset{˙}{\cdot} X$

Step	Hyp	Ref	Expression
1		ringcl.b	$⊢ B = {Base}_{R}$
2		ringcl.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
3		eqid	$⊢ {mulGrp}_{R} = {mulGrp}_{R}$
4	3	crngmgp	$⊢ R \in CRing \to {mulGrp}_{R} \in CMnd$
5	3 1	mgpbas	$⊢ B = {Base}_{{mulGrp}_{R}}$
6	3 2	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{{mulGrp}_{R}}$
7	5 6	cmncom	$⊢ ({mulGrp}_{R} \in CMnd \land X \in B \land Y \in B) \to X \underset{˙}{\cdot} Y = Y \underset{˙}{\cdot} X$
8	4 7	syl3an1	$⊢ (R \in CRing \land X \in B \land Y \in B) \to X \underset{˙}{\cdot} Y = Y \underset{˙}{\cdot} X$

Metamath Proof Explorer