Theorem ringsubdir

Description: Ring multiplication distributes over subtraction. ( subdir analog.) (Contributed by Jeff Madsen, 19-Jun-2010) (Revised by Mario Carneiro, 2-Jul-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ringsubdi.b	$$⊢B={\mathrm{Base}}_R$$
2		ringsubdi.t	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
3		ringsubdi.m	$$⊢\underset{\dot{}}{-}={-}_R$$
4		ringsubdi.r	$$⊢\phi \to R\in \mathrm{Ring}$$
5		ringsubdi.x	$$⊢\phi \to X\in B$$
6		ringsubdi.y	$$⊢\phi \to Y\in B$$
7		ringsubdi.z	$$⊢\phi \to Z\in B$$
8		ringrng	$$⊢R\in \mathrm{Ring}\to R\in \mathrm{Rng}$$
9	4 8	syl	$$⊢\phi \to R\in \mathrm{Rng}$$
10	1 2 3 9 5 6 7	rngsubdir	$$⊢\phi \to \left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{·}Z=\left(X\underset{\dot{}}{·}Z\right)\underset{\dot{}}{-}\left(Y\underset{\dot{}}{·}Z\right)$$