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Theorem adddiri 7784
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 7764 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
51, 2, 3, 4mp3an 1315 1  |-  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7625    + caddc 7630    x. cmul 7632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-addcl 7723  ax-mulcom 7728  ax-distr 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  numma  9232  binom2i  10408  3dvdsdec  11568  3dvds2dec  11569  sincosq3sgn  12925  sincosq4sgn  12926  cosq23lt0  12930
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