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Metamath Proof Explorer

Theorem ringsubdi

Description: Ring multiplication distributes over subtraction. ( subdi 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	ringsubdi	⊢ ( 𝜑 → ( 𝑋 · ( 𝑌 − 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) − ( 𝑋 · 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	rngsubdi	⊢ ( 𝜑 → ( 𝑋 · ( 𝑌 − 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) − ( 𝑋 · 𝑍 ) ) )