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Theorem adddiri 7553
Description: Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
Hypotheses
Ref Expression
axi.1  |-  A  e.  CC
axi.2  |-  B  e.  CC
axi.3  |-  C  e.  CC
Assertion
Ref Expression
adddiri  |-  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )

Proof of Theorem adddiri
StepHypRef Expression
1 axi.1 . 2  |-  A  e.  CC
2 axi.2 . 2  |-  B  e.  CC
3 axi.3 . 2  |-  C  e.  CC
4 adddir 7533 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
51, 2, 3, 4mp3an 1274 1  |-  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1290    e. wcel 1439  (class class class)co 5666   CCcc 7402    + caddc 7407    x. cmul 7409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-addcl 7495  ax-mulcom 7500  ax-distr 7503
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669
This theorem is referenced by:  numma  8974  binom2i  10117  3dvdsdec  11197  3dvds2dec  11198
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