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Theorem div2subap 9059
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 8420 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2 subcl 8420 . . . . 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 8833 . . . . . . . 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 8519 . . . . . . . 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 8519 . . . . . . . 8 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> -u D e. CC )
10 apadd2 8831 . . . . . . . 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 8522 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u C ) = 0 )
146, 8negsubd 8538 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> ( C + -u D ) = ( C - D ) )
1512, 13, 143brtr3d 4124 . . . . 5 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 # ( C - D ) )
16 0cnd 8215 . . . . . 6 |- ( ( C e. CC /\ D e. CC /\ C # D ) -> 0 e. CC )
17 apsym 8828 . . . . . 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 8957 . . . 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 8480 . . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
25 negsubdi2 8480 . . . 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 6047 . 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 2266 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 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8073   0cc0 8075   + caddc 8078   - cmin 8392   -ucneg 8393   # cap 8803   / cdiv 8894
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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895
This theorem is referenced by:  div2subapd  9060
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