Theorem rngass

Description: Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011) (Revised by AV, 13-Feb-2025)

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

Proof

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