ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  divdirap Unicode version
Theorem divdirap 8614
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 992 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2 simp2 993 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
3 recclap 8596 . . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
433ad2ant3 1015 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
51, 2, 4adddird 7945 . 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 7939 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A + B ) e. CC )
7 simp3l 1020 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
8 simp3r 1021 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
9 divrecap 8605 . . 3 |- ( ( ( A + B ) e. CC /\ C e. CC /\ C # 0 ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
106, 7, 8, 9syl3anc 1233 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
11 divrecap 8605 . . . 4 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
121, 7, 8, 11syl3anc 1233 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
13 divrecap 8605 . . . 4 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
142, 7, 8, 13syl3anc 1233 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
1512, 14oveq12d 5871 . 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 2213 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   # cap 8500   / cdiv 8589
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590
This theorem is referenced by:  muldivdirap  8624  divsubdirap  8625  divadddivap  8644  divdirapzi  8681  divdirapi  8686  divdirapd  8746  2halves  9107  halfaddsub  9112  zdivadd  9301  nneoor  9314  2tnp1ge0ge0  10257  flqdiv  10277  crim  10822  efival  11695  divgcdcoprm0  12055  ptolemy  13539
  Copyright terms: Public domain W3C validator