Theorem ringdid 33209

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

Hypotheses

Ref	Expression
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

Ref	Expression
ringdid	⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem ringdid

Step	Hyp	Ref	Expression
1		ringdid.r	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		ringdid.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		ringdid.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ringdid.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
5		ringdid.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
6		ringdid.p	. . 3 ⊢ + = (+g‘𝑅)
7		ringdid.m	. . 3 ⊢ · = (.r‘𝑅)
8	5, 6, 7	ringdi 20287	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
9	1, 2, 3, 4, 8	syl13anc 1372	1 ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  ._rcmulr 17312  Ringcrg 20260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-ring 20262

This theorem is referenced by:  ringdi22  33211  rloccring  33242