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Theorem divdirap 8164
Description: Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
divdirap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  /  C )  +  ( B  /  C ) ) )

Proof of Theorem divdirap
StepHypRef Expression
1 simp1 943 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  A  e.  CC )
2 simp2 944 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  B  e.  CC )
3 recclap 8146 . . . 4  |-  ( ( C  e.  CC  /\  C #  0 )  ->  (
1  /  C )  e.  CC )
433ad2ant3 966 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( 1  /  C
)  e.  CC )
51, 2, 4adddird 7513 . 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 7507 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  +  B
)  e.  CC )
7 simp3l 971 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C  e.  CC )
8 simp3r 972 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C #  0 )
9 divrecap 8155 . . 3  |-  ( ( ( A  +  B
)  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  (
( A  +  B
)  /  C )  =  ( ( A  +  B )  x.  ( 1  /  C
) ) )
106, 7, 8, 9syl3anc 1174 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  +  B )  x.  ( 1  /  C ) ) )
11 divrecap 8155 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  =  ( A  x.  (
1  /  C ) ) )
121, 7, 8, 11syl3anc 1174 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
13 divrecap 8155 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
142, 7, 8, 13syl3anc 1174 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1512, 14oveq12d 5670 . 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 2130 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  +  B )  /  C
)  =  ( ( A  /  C )  +  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    + caddc 7353    x. cmul 7355   # cap 8058    / cdiv 8139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140
This theorem is referenced by:  muldivdirap  8174  divsubdirap  8175  divadddivap  8194  divdirapzi  8231  divdirapi  8236  divdirapd  8296  2halves  8645  halfaddsub  8650  zdivadd  8835  nneoor  8848  2tnp1ge0ge0  9708  flqdiv  9728  crim  10292  efival  11023  divgcdcoprm0  11361
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