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Theorem div2subap 8457
Description: Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
Assertion
Ref Expression
div2subap  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C #  D
) )  ->  (
( A  -  B
)  /  ( C  -  D ) )  =  ( ( B  -  A )  / 
( D  -  C
) ) )

Proof of Theorem div2subap
StepHypRef Expression
1 subcl 7832 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2 subcl 7832 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  -  D
)  e.  CC )
323adant3 969 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  -  D )  e.  CC )
4 apneg 8239 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C #  D  <->  -u C #  -u D
) )
54biimp3a 1291 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u C #  -u D )
6 simp1 949 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  C  e.  CC )
76negcld 7931 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u C  e.  CC )
8 simp2 950 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  D  e.  CC )
98negcld 7931 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u D  e.  CC )
10 apadd2 8237 . . . . . . . 8  |-  ( (
-u C  e.  CC  /\  -u D  e.  CC  /\  C  e.  CC )  ->  ( -u C #  -u D  <->  ( C  +  -u C ) #  ( C  +  -u D ) ) )
117, 9, 6, 10syl3anc 1184 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( -u C #  -u D  <->  ( C  +  -u C ) #  ( C  +  -u D
) ) )
125, 11mpbid 146 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  +  -u C ) #  ( C  +  -u D ) )
136negidd 7934 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  +  -u C )  =  0 )
146, 8negsubd 7950 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  +  -u D )  =  ( C  -  D ) )
1512, 13, 143brtr3d 3904 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  0 #  ( C  -  D
) )
16 0cnd 7631 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  0  e.  CC )
17 apsym 8234 . . . . . 6  |-  ( ( 0  e.  CC  /\  ( C  -  D
)  e.  CC )  ->  ( 0 #  ( C  -  D )  <-> 
( C  -  D
) #  0 ) )
1816, 3, 17syl2anc 406 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  (
0 #  ( C  -  D )  <->  ( C  -  D ) #  0 ) )
1915, 18mpbid 146 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  -  D ) #  0 )
203, 19jca 302 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  (
( C  -  D
)  e.  CC  /\  ( C  -  D
) #  0 ) )
21 div2negap 8356 . . . 4  |-  ( ( ( A  -  B
)  e.  CC  /\  ( C  -  D
)  e.  CC  /\  ( C  -  D
) #  0 )  -> 
( -u ( A  -  B )  /  -u ( C  -  D )
)  =  ( ( A  -  B )  /  ( C  -  D ) ) )
22213expb 1150 . . 3  |-  ( ( ( A  -  B
)  e.  CC  /\  ( ( C  -  D )  e.  CC  /\  ( C  -  D
) #  0 ) )  ->  ( -u ( A  -  B )  /  -u ( C  -  D ) )  =  ( ( A  -  B )  /  ( C  -  D )
) )
231, 20, 22syl2an 285 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C #  D
) )  ->  ( -u ( A  -  B
)  /  -u ( C  -  D )
)  =  ( ( A  -  B )  /  ( C  -  D ) ) )
24 negsubdi2 7892 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
25 negsubdi2 7892 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  -> 
-u ( C  -  D )  =  ( D  -  C ) )
26253adant3 969 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u ( C  -  D )  =  ( D  -  C ) )
2724, 26oveqan12d 5725 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C #  D
) )  ->  ( -u ( A  -  B
)  /  -u ( C  -  D )
)  =  ( ( B  -  A )  /  ( D  -  C ) ) )
2823, 27eqtr3d 2134 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC  /\  C #  D
) )  ->  (
( A  -  B
)  /  ( C  -  D ) )  =  ( ( B  -  A )  / 
( D  -  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   0cc0 7500    + caddc 7503    - cmin 7804   -ucneg 7805   # cap 8209    / cdiv 8293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294
This theorem is referenced by:  div2subapd  8458
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