Theorem ringdi 20298

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 20286	. 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 498	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  +_gcplusg 17277  ._rcmulr 17278  Ringcrg 20270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394  df-ring 20272

This theorem is referenced by:  ringdid  20300  ringcomlem  20316  ringnegr  20340  ringlghm  20349  prdsringd  20356  imasring  20366  issubrg2  20629  cntzsubr  20643  sralmod  21242  psrlmod  21999  psrdi  22004  mamudir  22452  mdetrlin  22650  mdetuni0  22669  ply1divex  26185  erler  33407  rlocaddval  33411  lfladdcl  39656  lflvsdi2  39664  dvhlveclem  41693