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Theorem divdirap 8783
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 1000 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> A e. CC )
2 simp2 1001 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> B e. CC )
3 recclap 8765 . . . 4 |- ( ( C e. CC /\ C # 0 ) -> ( 1 / C ) e. CC )
433ad2ant3 1023 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( 1 / C ) e. CC )
51, 2, 4adddird 8111 . 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 8105 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A + B ) e. CC )
7 simp3l 1028 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C e. CC )
8 simp3r 1029 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> C # 0 )
9 divrecap 8774 . . 3 |- ( ( ( A + B ) e. CC /\ C e. CC /\ C # 0 ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
106, 7, 8, 9syl3anc 1250 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + B ) / C ) = ( ( A + B ) x. ( 1 / C ) ) )
11 divrecap 8774 . . . 4 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
121, 7, 8, 11syl3anc 1250 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
13 divrecap 8774 . . . 4 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
142, 7, 8, 13syl3anc 1250 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) = ( B x. ( 1 / C ) ) )
1512, 14oveq12d 5972 . 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 2249 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 104   /\ w3a 981   = wceq 1373   e. wcel 2177   class class class wbr 4048  (class class class)co 5954   CCcc 7936   0cc0 7938   1c1 7939   + caddc 7941   x. cmul 7943   # cap 8667   / cdiv 8758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-id 4345  df-po 4348  df-iso 4349  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-iota 5238  df-fun 5279  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759
This theorem is referenced by:  muldivdirap  8793  divsubdirap  8794  divadddivap  8813  divdirapzi  8850  divdirapi  8855  divdirapd  8915  2halves  9279  halfaddsub  9284  zdivadd  9475  nneoor  9488  2tnp1ge0ge0  10457  flqdiv  10479  mulsubdivbinom2ap  10869  crim  11219  efival  12093  divgcdcoprm0  12473  ptolemy  15346
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