Theorem ringass

Description: Associative law for multiplication in a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by NM, 27-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015)

	Ref	Expression
Hypotheses	ringcl.b	$$⊢B={\mathrm{Base}}_R$$
	ringcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	ringass	$$⊢\left(R\in \mathrm{Ring}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringcl.b	$$⊢B={\mathrm{Base}}_R$$
2		ringcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		eqid	$$⊢{\mathrm{mulGrp}}_R={\mathrm{mulGrp}}_R$$
4	3	ringmgp	$$⊢R\in \mathrm{Ring}\to {\mathrm{mulGrp}}_R\in \mathrm{Mnd}$$
5	3 1	mgpbas	$$⊢B={\mathrm{Base}}_{{\mathrm{mulGrp}}_R}$$
6	3 2	mgpplusg	$$⊢\underset{\dot{}}{·}={+}_{{\mathrm{mulGrp}}_R}$$
7	5 6	mndass	$$⊢\left({\mathrm{mulGrp}}_R\in \mathrm{Mnd}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$
8	4 7	sylan	$$⊢\left(R\in \mathrm{Ring}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$