Theorem ringdid

Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ringdid.b	$$⊢B={\mathrm{Base}}_R$$
2		ringdid.p	$$⊢\underset{\dot{}}{+}={+}_R$$
3		ringdid.m	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
4		ringdid.r	$$⊢\phi \to R\in \mathrm{Ring}$$
5		ringdid.x	$$⊢\phi \to X\in B$$
6		ringdid.y	$$⊢\phi \to Y\in B$$
7		ringdid.z	$$⊢\phi \to Z\in B$$
8	1 2 3	ringdi	$$⊢\left(R\in \mathrm{Ring}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{+}Z\right)=\left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{+}\left(X\underset{\dot{}}{·}Z\right)$$
9	4 5 6 7 8	syl13anc	$$⊢\phi \to X\underset{\dot{}}{·}\left(Y\underset{\dot{}}{+}Z\right)=\left(X\underset{\dot{}}{·}Y\right)\underset{\dot{}}{+}\left(X\underset{\dot{}}{·}Z\right)$$