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Theorem adddir 8917
Description: Distributive law for complex numbers. (Contributed by NM, 10-Oct-2004.)
Assertion
Ref Expression
adddir  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )

Proof of Theorem adddir
StepHypRef Expression
1 adddi 8913 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
213coml 1158 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
3 addcl 8906 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
4 mulcom 8910 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C
)  =  ( C  x.  ( A  +  B ) ) )
53, 4sylan 457 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C )  =  ( C  x.  ( A  +  B ) ) )
653impa 1146 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( C  x.  ( A  +  B
) ) )
7 mulcom 8910 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
873adant2 974 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 mulcom 8910 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 973 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
118, 10oveq12d 5960 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  +  ( B  x.  C ) )  =  ( ( C  x.  A )  +  ( C  x.  B
) ) )
122, 6, 113eqtr4d 2400 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710  (class class class)co 5942   CCcc 8822    + caddc 8827    x. cmul 8829
This theorem is referenced by:  mulid1  8922  adddiri  8935  adddird  8947  muladd11  9069  00id  9074  cnegex2  9081  muladd  9299  ser1const  11191  hashxplem  11475  demoivreALT  12572  dvds2ln  12650  dvds2add  12651  odd2np1lem  12677  cncrng  16495  icccvx  18546  sincosq1eq  19981  abssinper  19987  sineq0  19990  bposlem9  20637  cnrngo  21176  cncvc  21247  ipasslem1  21517  ipasslem11  21526  cdj3i  23129  expgrowth  26875  stoweidlem11  27083  stoweidlem13  27085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-addcl 8884  ax-mulcom 8888  ax-distr 8891
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945
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