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Theorem adddir 8830
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 8826 . . 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 8819 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
4 mulcom 8823 . . . 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 8823 . . . 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 8823 . . . 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 5876 . 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 2325 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 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742
This theorem is referenced by:  mulid1  8835  adddiri  8848  adddird  8860  muladd11  8982  00id  8987  cnegex2  8994  muladd  9212  ser1const  11102  hashxplem  11385  demoivreALT  12481  dvds2ln  12559  dvds2add  12560  odd2np1lem  12586  cncrng  16395  icccvx  18448  sincosq1eq  19880  abssinper  19886  sineq0  19889  bposlem9  20531  cnrngo  21070  cncvc  21139  ipasslem1  21409  ipasslem11  21418  cdj3i  23021  cnegvex2  25660  expgrowth  27552  stoweidlem11  27760  stoweidlem13  27762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-addcl 8797  ax-mulcom 8801  ax-distr 8804
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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