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Theorem divdir 10013
Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
divdir  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  /  C )  +  ( B  /  C ) ) )

Proof of Theorem divdir
StepHypRef Expression
1 simp1 983 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
2 simp2 984 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
3 reccl 9997 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 1  /  C
)  e.  CC )
433ad2ant3 1006 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  /  C
)  e.  CC )
51, 2, 4adddird 9407 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  x.  (
1  /  C ) )  =  ( ( A  x.  ( 1  /  C ) )  +  ( B  x.  ( 1  /  C
) ) ) )
61, 2addcld 9401 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  +  B
)  e.  CC )
7 simp3l 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
8 simp3r 1012 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  =/=  0 )
9 divrec 10006 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( A  +  B
)  /  C )  =  ( ( A  +  B )  x.  ( 1  /  C
) ) )
106, 7, 8, 9syl3anc 1213 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  +  B )  x.  ( 1  /  C ) ) )
11 divrec 10006 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  =  ( A  x.  (
1  /  C ) ) )
121, 7, 8, 11syl3anc 1213 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
13 divrec 10006 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
142, 7, 8, 13syl3anc 1213 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1512, 14oveq12d 6108 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  ( B  /  C ) )  =  ( ( A  x.  ( 1  /  C ) )  +  ( B  x.  ( 1  /  C
) ) ) )
165, 10, 153eqtr4d 2483 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  /  C )  +  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    / cdiv 9989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990
This theorem is referenced by:  divsubdir  10023  divadddiv  10042  divdirzi  10079  divdird  10141  2halves  10549  halfaddsub  10554  zdivadd  10709  nneo  10721  rpnnen1lem5  10979  fldiv  11695  modcyc  11739  crim  12600  efival  13432  pythagtriplem17  13894  ptolemy  21917  harmonicbnd4  22363  ppiub  22502  logfacrlim  22522  bposlem9  22590  chpchtlim  22687  mudivsum  22738  selberglem2  22754  pntrsumo1  22773  pntibndlem2  22799  pntibndlem3  22800  pntlemb  22805  heiborlem6  28640  sinhpcosh  30916  onetansqsecsq  30937  dpfrac1  30948
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