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Theorem div2subap 9132
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 8490 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2 subcl 8490 . . . . 5 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
323adant3 1044 . . . 4 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C - D ) e. CC )
4 apneg 8904 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC ) -> ( C # D <-> -u C # -u D ) )
54biimp3a 1382 . . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u C # -u D )
6 simp1 1024 . . . . . . . . 9 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> C e. CC )
76negcld 8589 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u C e. CC )
8 simp2 1025 . . . . . . . . 9 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> D e. CC )
98negcld 8589 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u D e. CC )
10 apadd2 8902 . . . . . . . 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 1274 . . . . . . 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 8592 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u C ) = 0 )
146, 8negsubd 8608 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u D ) = ( C - D ) )
1512, 13, 143brtr3d 4146 . . . . 5 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 # ( C - D ) )
16 0cnd 8284 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 e. CC )
17 apsym 8899 . . . . . 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 9030 . . . 4 |- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC /\ ( C - D ) # 0 ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( A - B ) / ( C - D ) ) )
22213expb 1231 . . 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 8550 . . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
25 negsubdi2 8550 . . . 4 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
26253adant3 1044 . . 3 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u ( C - D ) = ( D - C ) )
2724, 26oveqan12d 6078 . 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 2269 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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6059   CCcc 8142   0cc0 8144   + caddc 8147   - cmin 8462   -ucneg 8463   # cap 8874   / cdiv 8967
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-id 4420  df-po 4423  df-iso 4424  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-iota 5318  df-fun 5360  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968
This theorem is referenced by:  div2subapd  9133
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