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Theorem divsubdir 10019
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 9602 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 divdir 10009 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  +  -u B )  /  C )  =  ( ( A  /  C )  +  (
-u B  /  C
) ) )
31, 2syl3an2 1252 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  /  C )  +  ( -u B  /  C ) ) )
4 negsub 9649 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 6101 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
653adant3 1008 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
73, 6eqtr3d 2472 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  -  B
)  /  C ) )
8 divneg 10018 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
983expb 1188 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
1093adant1 1006 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C
)  =  ( -u B  /  C ) )
1110oveq2d 6102 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  +  ( -u B  /  C ) ) )
12 divcl 9992 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
13123expb 1188 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
14133adant2 1007 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
15 divcl 9992 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  e.  CC )
16153expb 1188 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( B  /  C )  e.  CC )
17163adant1 1006 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  e.  CC )
1814, 17negsubd 9717 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
1911, 18eqtr3d 2472 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
207, 19eqtr3d 2472 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 965    = wceq 1369    e. wcel 1756    =/= wne 2601  (class class class)co 6086   CCcc 9272   0cc0 9274    + caddc 9277    - cmin 9587   -ucneg 9588    / cdiv 9985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986
This theorem is referenced by:  divsubdird  10138  1mhlfehlf  10536  halfpm6th  10538  halfaddsub  10550  zeo  10719  quoremz  11686  quoremnn0ALT  11688  facndiv  12056  cos2bnd  13464  rpnnen2lem3  13491  rpnnen2lem11  13499  pythagtriplem15  13888  ovolscalem1  20976  sinq12gt0  21949  sincos6thpi  21957  ang180lem2  22186  log2cnv  22319  log2tlbnd  22320  basellem3  22400  ppiub  22523  logfacrlim  22543  logexprlim  22544  bposlem8  22610  chtppilimlem1  22702  vmadivsum  22711  rplogsumlem2  22714  rpvmasumlem  22716  rplogsum  22756  mulog2sumlem1  22763  selberg2lem  22779  selberg2  22780  selbergr  22797  pntlemr  22831  pntlemj  22832  ballotth  26889  subdivcomb1  27353  subdivcomb2  27354  bpoly3  28170  nndivsub  28273  heiborlem6  28686  areaquad  29563  lhe4.4ex1a  29574  stirlinglem10  29849
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