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Theorem adddiri 7997
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 7977 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
51, 2, 3, 4mp3an 1348 1  |-  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160  (class class class)co 5895   CCcc 7838    + caddc 7843    x. cmul 7845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcl 7936  ax-mulcom 7941  ax-distr 7944
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5898
This theorem is referenced by:  numma  9456  binom2i  10659  3dvdsdec  11901  3dvds2dec  11902  sincosq3sgn  14701  sincosq4sgn  14702  cosq23lt0  14706  2lgsoddprmlem3c  14910  2lgsoddprmlem3d  14911
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