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Theorem divsubdirap 8981
Description: Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
Assertion
Ref Expression
divsubdirap |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
Proof of Theorem divsubdirap
StepHypRef Expression
1 negcl 8472 . . . 4 |- ( B e. CC -> -u B e. CC )
2 divdirap 8970 . . . 4 |- ( ( A e. CC /\ -u B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
31, 2syl3an2 1308 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
4 negsub 8520 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
54oveq1d 6064 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
653adant3 1044 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
73, 6eqtr3d 2267 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A - B ) / C ) )
8 divnegap 8979 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> -u ( B / C ) = ( -u B / C ) )
983expb 1231 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1093adant1 1042 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1110oveq2d 6065 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) + ( -u B / C ) ) )
12 divclap 8951 . . . . . 6 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) e. CC )
13123expb 1231 . . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
14133adant2 1043 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
15 divclap 8951 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) e. CC )
16153expb 1231 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
17163adant1 1042 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
1814, 17negsubd 8589 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) - ( B / C ) ) )
1911, 18eqtr3d 2267 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A / C ) - ( B / C ) ) )
207, 19eqtr3d 2267 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8124   0cc0 8126   + caddc 8129   - cmin 8443   -ucneg 8444   # cap 8854   / cdiv 8945
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946
This theorem is referenced by:  divsubdirapd  9103  1mhlfehlf  9455  halfpm6th  9457  halfaddsub  9471  zeo  9682  mulsubdivbinom2ap  11072  cos2bnd  12442  sinq12gt0  15687  sincos6thpi  15699
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