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Theorem divsubdir 9456
Description: Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
Assertion
Ref Expression
divsubdir  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  -  B )  /  C
)  =  ( ( A  /  C )  -  ( B  /  C ) ) )

Proof of Theorem divsubdir
StepHypRef Expression
1 negcl 9052 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 divdir 9447 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  +  -u B )  /  C )  =  ( ( A  /  C )  +  (
-u B  /  C
) ) )
31, 2syl3an2 1216 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  /  C )  +  ( -u B  /  C ) ) )
4 negsub 9095 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
653adant3 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
73, 6eqtr3d 2317 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  -  B
)  /  C ) )
8 divneg 9455 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
983expb 1152 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
1093adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C
)  =  ( -u B  /  C ) )
1110oveq2d 5874 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  +  ( -u B  /  C ) ) )
12 divcl 9430 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
13123expb 1152 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
14133adant2 974 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
15 divcl 9430 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  e.  CC )
16153expb 1152 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( B  /  C )  e.  CC )
17163adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  e.  CC )
1814, 17negsubd 9163 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
1911, 18eqtr3d 2317 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
207, 19eqtr3d 2317 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740    - cmin 9037   -ucneg 9038    / cdiv 9423
This theorem is referenced by:  divsubdird  9575  1mhlfehlf  9934  halfpm6th  9936  halfaddsub  9945  zeo  10097  quoremz  10959  quoremnn0ALT  10961  facndiv  11301  cos2bnd  12468  rpnnen2lem3  12495  rpnnen2lem11  12503  pythagtriplem15  12882  ovolscalem1  18872  sinq12gt0  19875  sincos6thpi  19883  ang180lem2  20108  log2cnv  20240  log2tlbnd  20241  basellem3  20320  ppiub  20443  logfacrlim  20463  logexprlim  20464  bposlem8  20530  chtppilimlem1  20622  vmadivsum  20631  rplogsumlem2  20634  rpvmasumlem  20636  rplogsum  20676  mulog2sumlem1  20683  selberg2lem  20699  selberg2  20700  selbergr  20717  pntlemr  20751  pntlemj  20752  ballotth  23096  coinflippvt  23685  subdivcomb1  24090  subdivcomb2  24091  bpoly3  24793  nndivsub  24896  cntrset  25602  heiborlem6  26540  lhe4.4ex1a  27546  stirlinglem10  27832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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