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 = ( Base ‘ 𝑅 )
	ringsubdi.t	⊢ · = ( .r ‘ 𝑅 )
	ringsubdi.m	⊢ − = ( -g ‘ 𝑅 )
	ringsubdi.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringsubdi.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringsubdi.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	ringsubdi.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	ringsubdir	⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) − ( 𝑌 · 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringsubdi.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringsubdi.t	⊢ · = ( .r ‘ 𝑅 )
3		ringsubdi.m	⊢ − = ( -g ‘ 𝑅 )
4		ringsubdi.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		ringsubdi.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringsubdi.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		ringsubdi.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8		ringrng	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
9	4 8	syl	⊢ ( 𝜑 → 𝑅 ∈ Rng )
10	1 2 3 9 5 6 7	rngsubdir	⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) − ( 𝑌 · 𝑍 ) ) )