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Theorem rngdir 16650
Description: Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
Hypotheses
Ref Expression
rngdi.b  |-  B  =  ( Base `  R
)
rngdi.p  |-  .+  =  ( +g  `  R )
rngdi.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
rngdir  |-  ( ( 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 rngdir
StepHypRef Expression
1 rngdi.b . . 3  |-  B  =  ( Base `  R
)
2 rngdi.p . . 3  |-  .+  =  ( +g  `  R )
3 rngdi.t . . 3  |-  .x.  =  ( .r `  R )
41, 2, 3rngi 16643 . 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 965    = wceq 1369    e. wcel 1756   ` cfv 5411  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   Ringcrg 16631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-nul 4414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-iota 5374  df-fv 5419  df-ov 6089  df-rng 16633
This theorem is referenced by:  rngcom  16659  rnglz  16667  rngnegl  16671  rngsubdir  16677  mulgass2  16678  rngrghm  16680  prdsrngd  16690  imasrng  16697  opprrng  16709  issubrg2  16861  cntzsubr  16873  sralmod  17242  psrlmod  17446  psrdir  17454  evlslem1  17573  frlmphl  18175  mamudi  18276  mdetrlin  18378  dvrdir  26205  lflvscl  32473  lflvsdi1  32474  dvhlveclem  34504
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