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Theorem muladdi 7406
Description: Product of two sums. (Contributed by NM, 17-May-1999.)
Hypotheses
Ref Expression
mulm1.1  |-  A  e.  CC
mulneg.2  |-  B  e.  CC
subdi.3  |-  C  e.  CC
muladdi.4  |-  D  e.  CC
Assertion
Ref Expression
muladdi  |-  ( ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B
) )  +  ( ( A  x.  D
)  +  ( C  x.  B ) ) )

Proof of Theorem muladdi
StepHypRef Expression
1 mulm1.1 . 2  |-  A  e.  CC
2 mulneg.2 . 2  |-  B  e.  CC
3 subdi.3 . 2  |-  C  e.  CC
4 muladdi.4 . 2  |-  D  e.  CC
5 muladd 7381 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  x.  ( C  +  D )
)  =  ( ( ( A  x.  C
)  +  ( D  x.  B ) )  +  ( ( A  x.  D )  +  ( C  x.  B
) ) ) )
61, 2, 3, 4, 5mp4an 403 1  |-  ( ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B
) )  +  ( ( A  x.  D
)  +  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393  (class class class)co 5512   CCcc 6887    + caddc 6892    x. cmul 6894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-addcl 6980  ax-mulcl 6982  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-distr 6988
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by: (None)
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