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Theorem divdirap 9030
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 1028 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2 simp2 1029 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
3 recclap 9012 . . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
433ad2ant3 1051 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
51, 2, 4adddird 8352 . 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 8346 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A + B ) e. CC )
7 simp3l 1056 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
8 simp3r 1057 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
9 divrecap 9021 . . 3 |- ( ( ( A + B ) e. CC /\ C e. CC /\ C # 0 ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
106, 7, 8, 9syl3anc 1278 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
11 divrecap 9021 . . . 4 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
121, 7, 8, 11syl3anc 1278 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
13 divrecap 9021 . . . 4 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
142, 7, 8, 13syl3anc 1278 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
1512, 14oveq12d 6103 . 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 2281 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8178   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   # cap 8912   / cdiv 9005
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298
This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006
This theorem is used by:  muldivdirap  9040  divsubdirap  9041  divadddivap  9060  divdirapzi  9097  divdirapi  9102  divdirapd  9162  2halves  9539  halfaddsub  9544  zdivadd  9740  nneoor  9753  2tnp1ge0ge0  10750  flqdiv  10772  mulsubdivbinom2ap  11164  crim  11638  efival  12517  divgcdcoprm0  12897  ptolemy  15978  ppiqub  16215
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