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Theorem divsubdir 10015
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 9598 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 divdir 10005 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  +  -u B )  /  C )  =  ( ( A  /  C )  +  (
-u B  /  C
) ) )
31, 2syl3an2 1245 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  /  C )  +  ( -u B  /  C ) ) )
4 negsub 9645 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 6095 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
653adant3 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
73, 6eqtr3d 2467 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  -  B
)  /  C ) )
8 divneg 10014 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
983expb 1181 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
1093adant1 999 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C
)  =  ( -u B  /  C ) )
1110oveq2d 6096 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  +  ( -u B  /  C ) ) )
12 divcl 9988 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
13123expb 1181 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
14133adant2 1000 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
15 divcl 9988 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  e.  CC )
16153expb 1181 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( B  /  C )  e.  CC )
17163adant1 999 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  e.  CC )
1814, 17negsubd 9713 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
1911, 18eqtr3d 2467 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
207, 19eqtr3d 2467 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  -  B )  /  C
)  =  ( ( A  /  C )  -  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596  (class class class)co 6080   CCcc 9268   0cc0 9270    + caddc 9273    - cmin 9583   -ucneg 9584    / cdiv 9981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982
This theorem is referenced by:  divsubdird  10134  1mhlfehlf  10532  halfpm6th  10534  halfaddsub  10546  zeo  10715  quoremz  11678  quoremnn0ALT  11680  facndiv  12048  cos2bnd  13455  rpnnen2lem3  13482  rpnnen2lem11  13490  pythagtriplem15  13879  ovolscalem1  20838  sinq12gt0  21854  sincos6thpi  21862  ang180lem2  22091  log2cnv  22224  log2tlbnd  22225  basellem3  22305  ppiub  22428  logfacrlim  22448  logexprlim  22449  bposlem8  22515  chtppilimlem1  22607  vmadivsum  22616  rplogsumlem2  22619  rpvmasumlem  22621  rplogsum  22661  mulog2sumlem1  22668  selberg2lem  22684  selberg2  22685  selbergr  22702  pntlemr  22736  pntlemj  22737  ballotth  26768  subdivcomb1  27231  subdivcomb2  27232  bpoly3  28048  nndivsub  28151  heiborlem6  28559  areaquad  29437  lhe4.4ex1a  29448  stirlinglem10  29724
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