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Theorem divsubdirap 8665
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 8157 . . . 4 |- ( B e. CC -> -u B e. CC )
2 divdirap 8654 . . . 4 |- ( ( A e. CC /\ -u B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
31, 2syl3an2 1272 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A / C ) + ( -u B / C ) ) )
4 negsub 8205 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
54oveq1d 5890 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
653adant3 1017 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A + -u B ) / C ) = ( ( A - B ) / C ) )
73, 6eqtr3d 2212 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A - B ) / C ) )
8 divnegap 8663 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> -u ( B / C ) = ( -u B / C ) )
983expb 1204 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1093adant1 1015 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> -u ( B / C ) = ( -u B / C ) )
1110oveq2d 5891 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) + ( -u B / C ) ) )
12 divclap 8635 . . . . . 6 |- ( ( A e. CC /\ C e. CC /\ C # 0 ) -> ( A / C ) e. CC )
13123expb 1204 . . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
14133adant2 1016 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A / C ) e. CC )
15 divclap 8635 . . . . . 6 |- ( ( B e. CC /\ C e. CC /\ C # 0 ) -> ( B / C ) e. CC )
16153expb 1204 . . . . 5 |- ( ( B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
17163adant1 1015 . . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( B / C ) e. CC )
1814, 17negsubd 8274 . . 3 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + -u ( B / C ) ) = ( ( A / C ) - ( B / C ) ) )
1911, 18eqtr3d 2212 . 2 |- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( ( A / C ) + ( -u B / C ) ) = ( ( A / C ) - ( B / C ) ) )
207, 19eqtr3d 2212 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 978   = wceq 1353   e. wcel 2148   class class class wbr 4004  (class class class)co 5875   CCcc 7809   0cc0 7811   + caddc 7814   - cmin 8128   -ucneg 8129   # cap 8538   / cdiv 8629
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630
This theorem is referenced by:  divsubdirapd  8787  1mhlfehlf  9137  halfpm6th  9139  halfaddsub  9153  zeo  9358  mulsubdivbinom2ap  10691  cos2bnd  11768  sinq12gt0  14254  sincos6thpi  14266
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