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Theorem ringdi 17344
Description: Distributive law for the multiplication operation of a ring (left-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
ringdi  |-  ( ( 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 ringdi
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 17338 . 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 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   .rcmulr 14713   Ringcrg 17325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-ring 17327
This theorem is referenced by:  ringcom  17354  ringrz  17363  rngnegr  17368  ringsubdi  17372  ringlghm  17377  prdsringd  17388  imasring  17395  opprring  17407  issubrg2  17576  cntzsubr  17588  sralmod  17960  psrlmod  18181  psrdi  18188  mamudir  19033  mdetrlin  19231  mdetuni0  19250  ply1divex  22663  lidlrng  32877  lfladdcl  34939  lflvsdi2  34947  dvhlveclem  36978
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