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Theorem divdirap 8593
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 987 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2 simp2 988 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
3 recclap 8575 . . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
433ad2ant3 1010 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
51, 2, 4adddird 7924 . 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 7918 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A + B ) e. CC )
7 simp3l 1015 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
8 simp3r 1016 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
9 divrecap 8584 . . 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 divrecap 8584 . . . 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 divrecap 8584 . . . 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 5860 . 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 2208 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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   # cap 8479   / cdiv 8568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569
This theorem is referenced by:  muldivdirap  8603  divsubdirap  8604  divadddivap  8623  divdirapzi  8660  divdirapi  8665  divdirapd  8725  2halves  9086  halfaddsub  9091  zdivadd  9280  nneoor  9293  2tnp1ge0ge0  10236  flqdiv  10256  crim  10800  efival  11673  divgcdcoprm0  12033  ptolemy  13385
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