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Theorem divsubdir 10229
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 9809 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 divdir 10219 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  +  -u B )  /  C )  =  ( ( A  /  C )  +  (
-u B  /  C
) ) )
31, 2syl3an2 1257 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  /  C )  +  ( -u B  /  C ) ) )
4 negsub 9856 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
54oveq1d 6290 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
653adant3 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  +  -u B )  /  C
)  =  ( ( A  -  B )  /  C ) )
73, 6eqtr3d 2503 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  -  B
)  /  C ) )
8 divneg 10228 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
983expb 1192 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C )  =  ( -u B  /  C ) )
1093adant1 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  -u ( B  /  C
)  =  ( -u B  /  C ) )
1110oveq2d 6291 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  +  ( -u B  /  C ) ) )
12 divcl 10202 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
13123expb 1192 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
14133adant2 1010 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
15 divcl 10202 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  e.  CC )
16153expb 1192 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( B  /  C )  e.  CC )
17163adant1 1009 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  e.  CC )
1814, 17negsubd 9925 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  -u ( B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
1911, 18eqtr3d 2503 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  +  (
-u B  /  C
) )  =  ( ( A  /  C
)  -  ( B  /  C ) ) )
207, 19eqtr3d 2503 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655  (class class class)co 6275   CCcc 9479   0cc0 9481    + caddc 9484    - cmin 9794   -ucneg 9795    / cdiv 10195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196
This theorem is referenced by:  divsubdird  10348  1mhlfehlf  10747  halfpm6th  10749  halfaddsub  10761  zeo  10935  quoremz  11938  quoremnn0ALT  11940  facndiv  12321  cos2bnd  13773  rpnnen2lem3  13800  rpnnen2lem11  13808  pythagtriplem15  14201  ovolscalem1  21652  sinq12gt0  22626  sincos6thpi  22634  ang180lem2  22863  log2cnv  22996  log2tlbnd  22997  basellem3  23077  ppiub  23200  logfacrlim  23220  logexprlim  23221  bposlem8  23287  chtppilimlem1  23379  vmadivsum  23388  rplogsumlem2  23391  rpvmasumlem  23393  rplogsum  23433  mulog2sumlem1  23440  selberg2lem  23456  selberg2  23457  selbergr  23474  pntlemr  23508  pntlemj  23509  ballotth  27966  subdivcomb1  28430  subdivcomb2  28431  bpoly3  29247  nndivsub  29349  heiborlem6  29766  areaquad  30642  lhe4.4ex1a  30653  stirlinglem10  31202
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