Theorem 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	⊢ · = ( .r ‘ 𝑅 )
Assertion	crngcom	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringcl.t	⊢ · = ( .r ‘ 𝑅 )
3		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4	3	crngmgp	⊢ ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
5	3 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6	3 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7	5 6	cmncom	⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )
8	4 7	syl3an1	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )