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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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