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Theorem adddird 7576
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 7542 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
51, 2, 3, 4syl3anc 1175 1  |-  ( ph  ->  ( ( A  +  B )  x.  C
)  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439  (class class class)co 5668   CCcc 7411    + caddc 7416    x. cmul 7418
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 7504  ax-mulcom 7509  ax-distr 7512
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 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-iota 4995  df-fv 5038  df-ov 5671
This theorem is referenced by:  adddirp1d  7577  joinlmuladdmuld  7578  1p1times  7679  recextlem1  8183  divdirap  8227  subsq  10124  subsq2  10125  binom2  10128  binom3  10134  remullem  10368  resqrexlemover  10506  resqrexlemcalc1  10510  binomlem  10940
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