Theorem ringdi 20388

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

Hypotheses

Ref	Expression
ringdi.b	⊢ 𝐵 = (Base‘𝑅)
ringdi.p	⊢ + = (+g‘𝑅)
ringdi.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
ringdi	⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem ringdi

Step	Hyp	Ref	Expression
1		ringdi.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		ringdi.p	. . 3 ⊢ + = (+g‘𝑅)
3		ringdi.t	. . 3 ⊢ · = (.r‘𝑅)
4	1, 2, 3	ringdilem 20375	. 2 ⊢ ((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 500	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  ._rcmulr 17333  Ringcrg 20359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-ring 20361

This theorem is used by:  ringdid  20390  ringcomlem  20407  ringnegr  20432  ringlghm  20441  prdsringd  20448  imasring  20458  issubrg2  20741  cntzsubr  20755  sralmod  21358  psrlmod  22159  psrdi  22164  mamudir  22611  mdetrlin  22809  mdetuni0  22828  ply1divex  26345  erler  33649  rlocaddval  33653  lfladdcl  39903  lflvsdi2  39911  dvhlveclem  41940