MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ringdir Structured version   Visualization version   GIF version

Theorem ringdir 20156
Description: Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
Hypotheses
Ref Expression
ringdi.b 𝐵 = (Base‘𝑅)
ringdi.p + = (+g𝑅)
ringdi.t · = (.r𝑅)
Assertion
Ref Expression
ringdir ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Proof of Theorem ringdir
StepHypRef Expression
1 ringdi.b . . 3 𝐵 = (Base‘𝑅)
2 ringdi.p . . 3 + = (+g𝑅)
3 ringdi.t . . 3 · = (.r𝑅)
41, 2, 3ringdilem 20146 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
54simprd 495 1 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  .rcmulr 17199  Ringcrg 20130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-ring 20132
This theorem is referenced by:  ringo2times  20166  ringcomlem  20170  ringnegl  20193  mulgass2  20200  ringrghm  20204  prdsringd  20212  imasring  20221  dvrdir  20306  issubrg2  20486  cntzsubr  20500  sralmod  21035  frlmphl  21646  psrlmod  21833  psrdir  21839  evlslem1  21957  mamudi  22227  mdetrlin  22428  mxidlprm  33058  q1pdir  33142  r1pcyc  33146  lflvscl  38441  lflvsdi1  38442  dvhlveclem  40473
  Copyright terms: Public domain W3C validator