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Theorem divsubdivap 8716
Description: Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
divsubdivap  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  -  ( B  /  D
) )  =  ( ( ( A  x.  D )  -  ( B  x.  C )
)  /  ( C  x.  D ) ) )

Proof of Theorem divsubdivap
StepHypRef Expression
1 negcl 8188 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 divadddivap 8715 . . . 4  |-  ( ( ( A  e.  CC  /\  -u B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  +  ( -u B  /  D ) )  =  ( ( ( A  x.  D )  +  ( -u B  x.  C ) )  / 
( C  x.  D
) ) )
31, 2sylanl2 403 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  +  ( -u B  /  D ) )  =  ( ( ( A  x.  D )  +  ( -u B  x.  C ) )  / 
( C  x.  D
) ) )
4 simplr 528 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  B  e.  CC )
5 simprrl 539 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D  e.  CC )
6 simprrr 540 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D #  0 )
7 divnegap 8694 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  -u ( B  /  D )  =  ( -u B  /  D ) )
84, 5, 6, 7syl3anc 1249 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  -u ( B  /  D )  =  (
-u B  /  D
) )
98oveq2d 5913 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  + 
-u ( B  /  D ) )  =  ( ( A  /  C )  +  (
-u B  /  D
) ) )
10 simpll 527 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  A  e.  CC )
11 simprll 537 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C  e.  CC )
12 simprlr 538 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C #  0 )
13 divclap 8666 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
1410, 11, 12, 13syl3anc 1249 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  /  C )  e.  CC )
15 divclap 8666 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
164, 5, 6, 15syl3anc 1249 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  /  D )  e.  CC )
1714, 16negsubd 8305 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  + 
-u ( B  /  D ) )  =  ( ( A  /  C )  -  ( B  /  D ) ) )
189, 17eqtr3d 2224 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  +  ( -u B  /  D ) )  =  ( ( A  /  C )  -  ( B  /  D ) ) )
193, 18eqtr3d 2224 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  x.  D )  +  ( -u B  x.  C ) )  / 
( C  x.  D
) )  =  ( ( A  /  C
)  -  ( B  /  D ) ) )
204, 11mulneg1d 8399 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( -u B  x.  C )  =  -u ( B  x.  C
) )
2120oveq2d 5913 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  x.  D )  +  ( -u B  x.  C ) )  =  ( ( A  x.  D )  +  -u ( B  x.  C
) ) )
2210, 5mulcld 8009 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  x.  D )  e.  CC )
234, 11mulcld 8009 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  x.  C )  e.  CC )
2422, 23negsubd 8305 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  x.  D )  + 
-u ( B  x.  C ) )  =  ( ( A  x.  D )  -  ( B  x.  C )
) )
2521, 24eqtrd 2222 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  x.  D )  +  ( -u B  x.  C ) )  =  ( ( A  x.  D )  -  ( B  x.  C )
) )
2625oveq1d 5912 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  x.  D )  +  ( -u B  x.  C ) )  / 
( C  x.  D
) )  =  ( ( ( A  x.  D )  -  ( B  x.  C )
)  /  ( C  x.  D ) ) )
2719, 26eqtr3d 2224 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  -  ( B  /  D
) )  =  ( ( ( A  x.  D )  -  ( B  x.  C )
)  /  ( C  x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   CCcc 7840   0cc0 7842    + caddc 7845    x. cmul 7847    - cmin 8159   -ucneg 8160   # cap 8569    / cdiv 8660
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661
This theorem is referenced by:  subrecap  8827
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