Theorem ringdird 20349

Description: Distributive law for the multiplication operation of a ring (right-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
ringdird	⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Proof of Theorem ringdird

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	ringdir 20347	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
9	1, 2, 3, 4, 8	syl13anc 1397	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  ._rcmulr 17314  Ringcrg 20318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-12 2220  ax-ext 2742  ax-nul 5274

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6496  df-fv 6548  df-ov 7417  df-ring 20320

This theorem is referenced by:  ringdi22  20350  psdpw  22316  rloccring  33561  dflringlem2  33755  zrhcntr  34339