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Theorem rngdir 15360
Description: Distributive law for the multiplication operation of a ring. (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 15353 . 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 449 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   Ringcrg 15337
This theorem is referenced by:  rngcom  15369  rnglz  15377  rngnegl  15380  rngsubdir  15386  mulgass2  15387  rngrghm  15389  prdsrngd  15395  imasrng  15402  opprrng  15413  issubrg2  15565  cntzsubr  15577  sralmod  15939  psrlmod  16146  psrdir  16152  evlslem1  19399  mamudi  27461  lflvscl  29267  lflvsdi1  29268  dvhlveclem  31298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-rng 15340
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