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Theorem rngdi 17018
Description: Distributive law for the multiplication operation of a ring (left-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
rngdi  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  ( Y  .+  Z
) )  =  ( ( X  .x.  Y
)  .+  ( X  .x.  Z ) ) )

Proof of Theorem rngdi
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 17012 . 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 ) ) ) )
54simpld 459 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .x.  ( Y  .+  Z
) )  =  ( ( X  .x.  Y
)  .+  ( X  .x.  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   .rcmulr 14556   Ringcrg 17000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287  df-rng 17002
This theorem is referenced by:  rngcom  17028  rngrz  17037  rngnegr  17042  rngsubdi  17046  rnglghm  17051  prdsrngd  17062  imasrng  17069  opprrng  17081  issubrg2  17249  cntzsubr  17261  sralmod  17633  psrlmod  17853  psrdi  17860  mamudir  18701  mdetrlin  18899  mdetuni0  18918  ply1divex  22300  lfladdcl  33886  lflvsdi2  33894  dvhlveclem  35923
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