MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divdir Structured version   Unicode version

Theorem divdir 10022
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 988 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
2 simp2 989 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
3 reccl 10006 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 1  /  C
)  e.  CC )
433ad2ant3 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  /  C
)  e.  CC )
51, 2, 4adddird 9416 . 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 9410 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  +  B
)  e.  CC )
7 simp3l 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
8 simp3r 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  =/=  0 )
9 divrec 10015 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( A  +  B
)  /  C )  =  ( ( A  +  B )  x.  ( 1  /  C
) ) )
106, 7, 8, 9syl3anc 1218 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  +  B )  x.  ( 1  /  C ) ) )
11 divrec 10015 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  =  ( A  x.  (
1  /  C ) ) )
121, 7, 8, 11syl3anc 1218 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
13 divrec 10015 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
142, 7, 8, 13syl3anc 1218 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1512, 14oveq12d 6114 . 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 2485 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    / cdiv 9998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999
This theorem is referenced by:  divsubdir  10032  divadddiv  10051  divdirzi  10088  divdird  10150  2halves  10558  halfaddsub  10563  zdivadd  10718  nneo  10730  rpnnen1lem5  10988  fldiv  11704  modcyc  11748  crim  12609  efival  13441  pythagtriplem17  13903  ptolemy  21963  harmonicbnd4  22409  ppiub  22548  logfacrlim  22568  bposlem9  22636  chpchtlim  22733  mudivsum  22784  selberglem2  22800  pntrsumo1  22819  pntibndlem2  22845  pntibndlem3  22846  pntlemb  22851  heiborlem6  28720  sinhpcosh  31080  onetansqsecsq  31101  dpfrac1  31112
  Copyright terms: Public domain W3C validator