MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngdir Structured version   Unicode version

Theorem rngdir 15714
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 15707 . 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 451 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727   ` cfv 5483  (class class class)co 6110   Basecbs 13500   +g cplusg 13560   .rcmulr 13561   Ringcrg 15691
This theorem is referenced by:  rngcom  15723  rnglz  15731  rngnegl  15734  rngsubdir  15740  mulgass2  15741  rngrghm  15743  prdsrngd  15749  imasrng  15756  opprrng  15767  issubrg2  15919  cntzsubr  15931  sralmod  16289  psrlmod  16496  psrdir  16502  evlslem1  19967  dvrdir  24257  mamudi  27476  lflvscl  29973  lflvsdi1  29974  dvhlveclem  32004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-nul 4363
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-ov 6113  df-rng 15694
  Copyright terms: Public domain W3C validator