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Theorem ringdir 17196
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  |-  .+  =  ( +g  `  R )
ringdi.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
ringdir  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) )

Proof of Theorem ringdir
StepHypRef Expression
1 ringdi.b . . 3  |-  B  =  ( Base `  R
)
2 ringdi.p . . 3  |-  .+  =  ( +g  `  R )
3 ringdi.t . . 3  |-  .x.  =  ( .r `  R )
41, 2, 3ringi 17189 . 2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .x.  ( Y  .+  Z ) )  =  ( ( X  .x.  Y )  .+  ( X  .x.  Z ) )  /\  ( ( X 
.+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) ) )
54simprd 463 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Y )  .x.  Z )  =  ( ( X  .x.  Z
)  .+  ( Y  .x.  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14613   +g cplusg 14678   .rcmulr 14679   Ringcrg 17176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-ring 17178
This theorem is referenced by:  ringcom  17205  ringlz  17213  ringnegl  17218  rngsubdir  17224  mulgass2  17225  ringrghm  17229  prdsringd  17239  imasring  17246  opprring  17258  issubrg2  17427  cntzsubr  17439  sralmod  17811  psrlmod  18032  psrdir  18040  evlslem1  18162  frlmphl  18789  mamudi  18882  mdetrlin  19081  dvrdir  27757  lidlrng  32443  lflvscl  34542  lflvsdi1  34543  dvhlveclem  36575
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