Metamath Proof Explorer

Theorem ringdid

Description: Distributive law for the multiplication operation of a ring (left-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 ringdid ( 𝜑 → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 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 ringdi ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )
9 4 5 6 7 8 syl13anc ⊢ ( 𝜑 → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )