Theorem ringdi

Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007)

	Ref	Expression
Hypotheses	ringdi.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringdi.p	⊢ + = ( +g ‘ 𝑅 )
	ringdi.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	ringdi	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringdi.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringdi.p	⊢ + = ( +g ‘ 𝑅 )
3		ringdi.t	⊢ · = ( .r ‘ 𝑅 )
4	1 2 3	ringdilem	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) ∧ ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) ) )
5	4	simpld	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )