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Theorem adddir 9085
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 9081 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
213coml 1161 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
3 addcl 9074 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
4 mulcom 9078 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C
)  =  ( C  x.  ( A  +  B ) ) )
53, 4sylan 459 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C )  =  ( C  x.  ( A  +  B ) ) )
653impa 1149 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( C  x.  ( A  +  B
) ) )
7 mulcom 9078 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
873adant2 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 mulcom 9078 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
118, 10oveq12d 6101 . 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 2480 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726  (class class class)co 6083   CCcc 8990    + caddc 8995    x. cmul 8997
This theorem is referenced by:  mulid1  9090  adddiri  9103  adddird  9115  muladd11  9238  00id  9243  cnegex2  9250  muladd  9468  ser1const  11381  hashxplem  11698  demoivreALT  12804  dvds2ln  12882  dvds2add  12883  odd2np1lem  12909  cncrng  16724  icccvx  18977  sincosq1eq  20422  abssinper  20428  sineq0  20431  bposlem9  21078  cnrngo  21993  cncvc  22064  ipasslem1  22334  ipasslem11  22343  cdj3i  23946  mblfinlem3  26247  expgrowth  27531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-addcl 9052  ax-mulcom 9056  ax-distr 9059
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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