Theorem crngmgp

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

	Ref	Expression
Hypothesis	ringmgp.g	$$⊢G={\mathrm{mulGrp}}_R$$
Assertion	crngmgp	$$⊢R\in \mathrm{CRing}\to G\in \mathrm{CMnd}$$

Proof

Step	Hyp	Ref	Expression
1		ringmgp.g	$$⊢G={\mathrm{mulGrp}}_R$$
2	1	iscrng	$$⊢R\in \mathrm{CRing}\leftrightarrow \left(R\in \mathrm{Ring}\wedge G\in \mathrm{CMnd}\right)$$
3	2	simprbi	$$⊢R\in \mathrm{CRing}\to G\in \mathrm{CMnd}$$