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Theorem cevathlem2 31881
Description: Ceva's theorem second lemma. Relate (doubled) areas of triangles  C A O and 
A B O with of segments  B D and 
D C. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
Hypotheses
Ref Expression
cevath.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
cevath.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
cevath.b  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
cevath.c  |-  ( ph  ->  O  e.  CC )
cevath.d  |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O
) )  =  0  /\  ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0 ) )
cevath.e  |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0 ) )
cevath.f  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
Assertion
Ref Expression
cevathlem2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y   
x, O, y    x, E, y    x, F, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem cevathlem2
StepHypRef Expression
1 cevath.sigar . . . . . . 7  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
2 cevath.b . . . . . . . . 9  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
32simp2d 1009 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
4 cevath.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
54simp1d 1008 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
64simp2d 1009 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
73, 5, 63jca 1176 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
8 cevath.c . . . . . . . 8  |-  ( ph  ->  O  e.  CC )
95, 8subcld 9942 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  O
)  e.  CC )
103, 8subcld 9942 . . . . . . . . . 10  |-  ( ph  ->  ( D  -  O
)  e.  CC )
119, 10jca 532 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( D  -  O
)  e.  CC ) )
12 cevath.d . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O
) )  =  0  /\  ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0 ) )
1312simp1d 1008 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
141, 11, 13sigariz 31876 . . . . . . . 8  |-  ( ph  ->  ( ( D  -  O ) G ( A  -  O ) )  =  0 )
158, 14jca 532 . . . . . . 7  |-  ( ph  ->  ( O  e.  CC  /\  ( ( D  -  O ) G ( A  -  O ) )  =  0 ) )
161, 7, 15sigaradd 31879 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
171sigarperm 31873 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( B  -  O
) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B
) ) )
186, 5, 8, 17syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
1916, 18eqtr4d 2511 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( B  -  O ) G ( A  -  O ) ) )
2019oveq1d 6310 . . . 4  |-  ( ph  ->  ( ( ( ( A  -  B ) G ( D  -  B ) )  -  ( ( O  -  B ) G ( D  -  B ) ) )  x.  ( C  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
215, 6subcld 9942 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
223, 6subcld 9942 . . . . . . 7  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2321, 22jca 532 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
241, 23sigarimcd 31875 . . . . 5  |-  ( ph  ->  ( ( A  -  B ) G ( D  -  B ) )  e.  CC )
258, 6subcld 9942 . . . . . . 7  |-  ( ph  ->  ( O  -  B
)  e.  CC )
2625, 22jca 532 . . . . . 6  |-  ( ph  ->  ( ( O  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
271, 26sigarimcd 31875 . . . . 5  |-  ( ph  ->  ( ( O  -  B ) G ( D  -  B ) )  e.  CC )
284simp3d 1010 . . . . . 6  |-  ( ph  ->  C  e.  CC )
2928, 3subcld 9942 . . . . 5  |-  ( ph  ->  ( C  -  D
)  e.  CC )
3024, 27, 29subdird 10025 . . . 4  |-  ( ph  ->  ( ( ( ( A  -  B ) G ( D  -  B ) )  -  ( ( O  -  B ) G ( D  -  B ) ) )  x.  ( C  -  D )
)  =  ( ( ( ( A  -  B ) G ( D  -  B ) )  x.  ( C  -  D ) )  -  ( ( ( O  -  B ) G ( D  -  B ) )  x.  ( C  -  D
) ) ) )
3120, 30eqtr3d 2510 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( A  -  O
) )  x.  ( C  -  D )
)  =  ( ( ( ( A  -  B ) G ( D  -  B ) )  x.  ( C  -  D ) )  -  ( ( ( O  -  B ) G ( D  -  B ) )  x.  ( C  -  D
) ) ) )
326, 28, 53jca 1176 . . . . 5  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
33 cevath.e . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0 ) )
3433simp2d 1009 . . . . . 6  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
353, 34jca 532 . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0 ) )
361, 32, 35sharhght 31878 . . . 4  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( A  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
376, 28, 83jca 1176 . . . . 5  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  O  e.  CC )
)
381, 37, 35sharhght 31878 . . . 4  |-  ( ph  ->  ( ( ( O  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( O  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
3936, 38oveq12d 6313 . . 3  |-  ( ph  ->  ( ( ( ( A  -  B ) G ( D  -  B ) )  x.  ( C  -  D
) )  -  (
( ( O  -  B ) G ( D  -  B ) )  x.  ( C  -  D ) ) )  =  ( ( ( ( A  -  C ) G ( D  -  C ) )  x.  ( B  -  D ) )  -  ( ( ( O  -  C ) G ( D  -  C ) )  x.  ( B  -  D
) ) ) )
405, 28subcld 9942 . . . . . . 7  |-  ( ph  ->  ( A  -  C
)  e.  CC )
413, 28subcld 9942 . . . . . . 7  |-  ( ph  ->  ( D  -  C
)  e.  CC )
421sigarim 31864 . . . . . . 7  |-  ( ( ( A  -  C
)  e.  CC  /\  ( D  -  C
)  e.  CC )  ->  ( ( A  -  C ) G ( D  -  C
) )  e.  RR )
4340, 41, 42syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  RR )
4443recnd 9634 . . . . 5  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  CC )
458, 28subcld 9942 . . . . . . 7  |-  ( ph  ->  ( O  -  C
)  e.  CC )
4645, 41jca 532 . . . . . 6  |-  ( ph  ->  ( ( O  -  C )  e.  CC  /\  ( D  -  C
)  e.  CC ) )
471, 46sigarimcd 31875 . . . . 5  |-  ( ph  ->  ( ( O  -  C ) G ( D  -  C ) )  e.  CC )
486, 3subcld 9942 . . . . 5  |-  ( ph  ->  ( B  -  D
)  e.  CC )
4944, 47, 48subdird 10025 . . . 4  |-  ( ph  ->  ( ( ( ( A  -  C ) G ( D  -  C ) )  -  ( ( O  -  C ) G ( D  -  C ) ) )  x.  ( B  -  D )
)  =  ( ( ( ( A  -  C ) G ( D  -  C ) )  x.  ( B  -  D ) )  -  ( ( ( O  -  C ) G ( D  -  C ) )  x.  ( B  -  D
) ) ) )
503, 5, 283jca 1176 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  C  e.  CC )
)
511, 50, 15sigaradd 31879 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
521sigarperm 31873 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( C  -  O
) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C
) ) )
5328, 5, 8, 52syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
5451, 53eqtr4d 2511 . . . . 5  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( C  -  O ) G ( A  -  O ) ) )
5554oveq1d 6310 . . . 4  |-  ( ph  ->  ( ( ( ( A  -  C ) G ( D  -  C ) )  -  ( ( O  -  C ) G ( D  -  C ) ) )  x.  ( B  -  D )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( B  -  D ) ) )
5649, 55eqtr3d 2510 . . 3  |-  ( ph  ->  ( ( ( ( A  -  C ) G ( D  -  C ) )  x.  ( B  -  D
) )  -  (
( ( O  -  C ) G ( D  -  C ) )  x.  ( B  -  D ) ) )  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( B  -  D ) ) )
5731, 39, 563eqtrrd 2513 . 2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
586, 8subcld 9942 . . . 4  |-  ( ph  ->  ( B  -  O
)  e.  CC )
591sigarac 31865 . . . 4  |-  ( ( ( B  -  O
)  e.  CC  /\  ( A  -  O
)  e.  CC )  ->  ( ( B  -  O ) G ( A  -  O
) )  =  -u ( ( A  -  O ) G ( B  -  O ) ) )
6058, 9, 59syl2anc 661 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  -u (
( A  -  O
) G ( B  -  O ) ) )
6160oveq1d 6310 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( A  -  O
) )  x.  ( C  -  D )
)  =  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D ) ) )
629, 58jca 532 . . . . 5  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
631, 62sigarimcd 31875 . . . 4  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
64 mulneg12 10007 . . . 4  |-  ( ( ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  D
)  e.  CC )  ->  ( -u (
( A  -  O
) G ( B  -  O ) )  x.  ( C  -  D ) )  =  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
) )
6563, 29, 64syl2anc 661 . . 3  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  -u ( C  -  D )
) )
6628, 3negsubdi2d 9958 . . . 4  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
6766oveq2d 6311 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6865, 67eqtrd 2508 . 2  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6957, 61, 683eqtrd 2512 1  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    - cmin 9817   -ucneg 9818   *ccj 12908   Imcim 12910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-cj 12911  df-re 12912  df-im 12913
This theorem is referenced by:  cevath  31882
  Copyright terms: Public domain W3C validator