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Theorem adddird 7798
Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addcld.1  |-  ( ph  ->  A  e.  CC )
addcld.2  |-  ( ph  ->  B  e.  CC )
addassd.3  |-  ( ph  ->  C  e.  CC )
Assertion
Ref Expression
adddird  |-  ( ph  ->  ( ( A  +  B )  x.  C
)  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )

Proof of Theorem adddird
StepHypRef Expression
1 addcld.1 . 2  |-  ( ph  ->  A  e.  CC )
2 addcld.2 . 2  |-  ( ph  ->  B  e.  CC )
3 addassd.3 . 2  |-  ( ph  ->  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, 4syl3anc 1216 1  |-  ( ph  ->  ( ( A  +  B )  x.  C
)  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  adddirp1d  7799  joinlmuladdmuld  7800  1p1times  7903  recextlem1  8419  divdirap  8464  subsq  10406  subsq2  10407  binom2  10410  binom3  10416  remullem  10650  resqrexlemover  10789  resqrexlemcalc1  10793  bdtrilem  11017  binomlem  11259  dvexp  12854
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