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Theorem ringdir 18921
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, 3ringi 18914 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
54simprd 491 1 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  .rcmulr 16306  Ringcrg 18901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-ring 18903
This theorem is referenced by:  ringadd2  18929  rngo2times  18930  ringcom  18933  ringlz  18941  ringnegl  18948  rngsubdir  18954  mulgass2  18955  ringrghm  18959  prdsringd  18966  imasring  18973  opprring  18985  issubrg2  19156  cntzsubr  19168  sralmod  19548  psrlmod  19762  psrdir  19768  evlslem1  19875  frlmphl  20487  mamudi  20576  mdetrlin  20776  dvrdir  30335  lflvscl  35152  lflvsdi1  35153  dvhlveclem  37183  lidlrng  42774
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