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Theorem div2subap 8603
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 7968 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2 subcl 7968 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  -  D
)  e.  CC )
323adant3 1001 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  -  D )  e.  CC )
4 apneg 8380 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C #  D  <->  -u C #  -u D
) )
54biimp3a 1323 . . . . . . 7  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u C #  -u D )
6 simp1 981 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  C  e.  CC )
76negcld 8067 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u C  e.  CC )
8 simp2 982 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  D  e.  CC )
98negcld 8067 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u D  e.  CC )
10 apadd2 8378 . . . . . . . 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 1216 . . . . . . 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 8070 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  +  -u C )  =  0 )
146, 8negsubd 8086 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  ( C  +  -u D )  =  ( C  -  D ) )
1512, 13, 143brtr3d 3959 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  0 #  ( C  -  D
) )
16 0cnd 7766 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  0  e.  CC )
17 apsym 8375 . . . . . 6  |-  ( ( 0  e.  CC  /\  ( C  -  D
)  e.  CC )  ->  ( 0 #  ( C  -  D )  <-> 
( C  -  D
) #  0 ) )
1816, 3, 17syl2anc 408 . . . . 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 304 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  (
( C  -  D
)  e.  CC  /\  ( C  -  D
) #  0 ) )
21 div2negap 8502 . . . 4  |-  ( ( ( A  -  B
)  e.  CC  /\  ( C  -  D
)  e.  CC  /\  ( C  -  D
) #  0 )  -> 
( -u ( A  -  B )  /  -u ( C  -  D )
)  =  ( ( A  -  B )  /  ( C  -  D ) ) )
22213expb 1182 . . 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 287 . 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 8028 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( A  -  B )  =  ( B  -  A ) )
25 negsubdi2 8028 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  -> 
-u ( C  -  D )  =  ( D  -  C ) )
26253adant3 1001 . . 3  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  C #  D )  ->  -u ( C  -  D )  =  ( D  -  C ) )
2724, 26oveqan12d 5793 . 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 2174 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 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625   0cc0 7627    + caddc 7630    - cmin 7940   -ucneg 7941   # cap 8350    / cdiv 8439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440
This theorem is referenced by:  div2subapd  8604
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