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Theorem divdir 10226
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 996 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  A  e.  CC )
2 simp2 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
3 reccl 10210 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 1  /  C
)  e.  CC )
433ad2ant3 1019 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( 1  /  C
)  e.  CC )
51, 2, 4adddird 9617 . 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 9611 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  +  B
)  e.  CC )
7 simp3l 1024 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
8 simp3r 1025 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  =/=  0 )
9 divrec 10219 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( A  +  B
)  /  C )  =  ( ( A  +  B )  x.  ( 1  /  C
) ) )
106, 7, 8, 9syl3anc 1228 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  +  B )  x.  ( 1  /  C ) ) )
11 divrec 10219 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  =  ( A  x.  (
1  /  C ) ) )
121, 7, 8, 11syl3anc 1228 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
13 divrec 10219 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
142, 7, 8, 13syl3anc 1228 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1512, 14oveq12d 6300 . 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 2518 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    / cdiv 10202
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  divsubdir  10236  divadddiv  10255  divdirzi  10292  divdird  10354  2halves  10763  halfaddsub  10768  zdivadd  10928  nneo  10940  rpnnen1lem5  11208  fldiv  11951  modcyc  11995  crim  12907  efival  13744  pythagtriplem17  14210  ptolemy  22622  harmonicbnd4  23068  ppiub  23207  logfacrlim  23227  bposlem9  23295  chpchtlim  23392  mudivsum  23443  selberglem2  23459  pntrsumo1  23478  pntibndlem2  23504  pntibndlem3  23505  pntlemb  23510  heiborlem6  29915  sinhpcosh  32215  onetansqsecsq  32236  dpfrac1  32247
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