Theorem ringassd

Description: Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024)

	Ref	Expression
Hypotheses	ringassd.b	$$⊢B={\mathrm{Base}}_R$$
	ringassd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	ringassd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
	ringassd.x	$$⊢\phi \to X\in B$$
	ringassd.y	$$⊢\phi \to Y\in B$$
	ringassd.z	$$⊢\phi \to Z\in B$$
Assertion	ringassd	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringassd.b	$$⊢B={\mathrm{Base}}_R$$
2		ringassd.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringassd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
4		ringassd.x	$$⊢\phi \to X\in B$$
5		ringassd.y	$$⊢\phi \to Y\in B$$
6		ringassd.z	$$⊢\phi \to Z\in B$$
7	1 2	ringass	$$⊢\left(R\in \mathrm{Ring}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$
8	3 4 5 6 7	syl13anc	$$⊢\phi \to \left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{·}Z=X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{·}Z\right)$$