ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  div2subap Unicode version
Theorem div2subap 8808
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 8170 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2 subcl 8170 . . . . 5 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
323adant3 1018 . . . 4 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C - D ) e. CC )
4 apneg 8582 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC ) -> ( C # D <-> -u C # -u D ) )
54biimp3a 1355 . . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u C # -u D )
6 simp1 998 . . . . . . . . 9 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> C e. CC )
76negcld 8269 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u C e. CC )
8 simp2 999 . . . . . . . . 9 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> D e. CC )
98negcld 8269 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u D e. CC )
10 apadd2 8580 . . . . . . . 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 1248 . . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( -u C # -u D <-> ( C + -u C ) # ( C + -u D ) ) )
125, 11mpbid 147 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u C ) # ( C + -u D ) )
136negidd 8272 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u C ) = 0 )
146, 8negsubd 8288 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u D ) = ( C - D ) )
1512, 13, 143brtr3d 4046 . . . . 5 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 # ( C - D ) )
16 0cnd 7964 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 e. CC )
17 apsym 8577 . . . . . 6 |- ( ( 0 e. CC /\ ( C - D ) e. CC ) -> ( 0 # ( C - D ) <-> ( C - D ) # 0 ) )
1816, 3, 17syl2anc 411 . . . . 5 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( 0 # ( C - D ) <-> ( C - D ) # 0 ) )
1915, 18mpbid 147 . . . 4 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C - D ) # 0 )
203, 19jca 306 . . 3 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( ( C - D ) e. CC /\ ( C - D ) # 0 ) )
21 div2negap 8706 . . . 4 |- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC /\ ( C - D ) # 0 ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( A - B ) / ( C - D ) ) )
22213expb 1205 . . 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 289 . 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 8230 . . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
25 negsubdi2 8230 . . . 4 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
26253adant3 1018 . . 3 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u ( C - D ) = ( D - C ) )
2724, 26oveqan12d 5907 . 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 2222 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 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7823   0cc0 7825   + caddc 7828   - cmin 8142   -ucneg 8143   # cap 8552   / cdiv 8643
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644
This theorem is referenced by:  div2subapd  8809
  Copyright terms: Public domain W3C validator