Theorem ringdird

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

	Ref	Expression
Hypotheses	ringdid.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringdid.p	⊢ + = ( +g ‘ 𝑅 )
	ringdid.m	⊢ · = ( .r ‘ 𝑅 )
	ringdid.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
	ringdid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringdid.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	ringdid.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	ringdird	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringdid.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringdid.p	⊢ + = ( +g ‘ 𝑅 )
3		ringdid.m	⊢ · = ( .r ‘ 𝑅 )
4		ringdid.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
5		ringdid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringdid.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		ringdid.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8	1 2 3	ringdir	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) )
9	4 5 6 7 8	syl13anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) )