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Theorem div2subap 8599
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 7964 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2 subcl 7964 . . . . 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 8376 . . . . . . . 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 8063 . . . . . . . 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 8063 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u D e. CC )
10 apadd2 8374 . . . . . . . 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 8066 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u C ) = 0 )
146, 8negsubd 8082 . . . . . 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 7762 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 e. CC )
17 apsym 8371 . . . . . 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 8498 . . . 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 8024 . . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
25 negsubdi2 8024 . . . 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 7621   0cc0 7623   + caddc 7626   - cmin 7936   -ucneg 7937   # cap 8346   / cdiv 8435
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 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741
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 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436
This theorem is referenced by:  div2subapd  8600
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