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Theorem adddir 9599
Description: Distributive law for complex numbers (right-distributivity). (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 9593 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
213coml 1203 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  ( A  +  B ) )  =  ( ( C  x.  A )  +  ( C  x.  B ) ) )
3 addcl 9586 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
4 mulcom 9590 . . . 4  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C
)  =  ( C  x.  ( A  +  B ) ) )
53, 4sylan 471 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C )  =  ( C  x.  ( A  +  B ) ) )
653impa 1191 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( C  x.  ( A  +  B
) ) )
7 mulcom 9590 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
873adant2 1015 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 mulcom 9590 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 1014 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
118, 10oveq12d 6313 . 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 2518 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6295   CCcc 9502    + caddc 9507    x. cmul 9509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-addcl 9564  ax-mulcom 9568  ax-distr 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  mulid1  9605  adddiri  9619  adddird  9633  muladd11  9761  00id  9766  cnegex2  9773  muladd  10001  ser1const  12143  hashxplem  12472  demoivreALT  13814  dvds2ln  13892  dvds2add  13893  odd2np1lem  13921  cncrng  18309  icccvx  21318  sincosq1eq  22771  abssinper  22777  sineq0  22780  bposlem9  23433  cnrngo  25228  cncvc  25299  ipasslem1  25569  ipasslem11  25578  cdj3i  27183  mblfinlem3  29980  expgrowth  31164  2zrngALT  32348
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