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Theorem cevathlem2 27789
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 970 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
4 cevath.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
54simp1d 969 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
64simp2d 970 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
73, 5, 63jca 1134 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
8 cevath.c . . . . . . . 8  |-  ( ph  ->  O  e.  CC )
95, 8subcld 9401 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  O
)  e.  CC )
103, 8subcld 9401 . . . . . . . . . 10  |-  ( ph  ->  ( D  -  O
)  e.  CC )
119, 10jca 519 . . . . . . . . 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 969 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
141, 11, 13sigariz 27784 . . . . . . . 8  |-  ( ph  ->  ( ( D  -  O ) G ( A  -  O ) )  =  0 )
158, 14jca 519 . . . . . . 7  |-  ( ph  ->  ( O  e.  CC  /\  ( ( D  -  O ) G ( A  -  O ) )  =  0 ) )
161, 7, 15sigaradd 27787 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
171sigarperm 27781 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( B  -  O
) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B
) ) )
186, 5, 8, 17syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
1916, 18eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( B  -  O ) G ( A  -  O ) ) )
2019oveq1d 6088 . . . 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 9401 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
223, 6subcld 9401 . . . . . . 7  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2321, 22jca 519 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
241, 23sigarimcd 27783 . . . . 5  |-  ( ph  ->  ( ( A  -  B ) G ( D  -  B ) )  e.  CC )
258, 6subcld 9401 . . . . . . 7  |-  ( ph  ->  ( O  -  B
)  e.  CC )
2625, 22jca 519 . . . . . 6  |-  ( ph  ->  ( ( O  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
271, 26sigarimcd 27783 . . . . 5  |-  ( ph  ->  ( ( O  -  B ) G ( D  -  B ) )  e.  CC )
284simp3d 971 . . . . . 6  |-  ( ph  ->  C  e.  CC )
2928, 3subcld 9401 . . . . 5  |-  ( ph  ->  ( C  -  D
)  e.  CC )
3024, 27, 29subdird 9480 . . . 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 2469 . . 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 1134 . . . . 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 970 . . . . . 6  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
353, 34jca 519 . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0 ) )
361, 32, 35sharhght 27786 . . . 4  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( A  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
376, 28, 83jca 1134 . . . . 5  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  O  e.  CC )
)
381, 37, 35sharhght 27786 . . . 4  |-  ( ph  ->  ( ( ( O  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( O  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
3936, 38oveq12d 6091 . . 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 9401 . . . . . . 7  |-  ( ph  ->  ( A  -  C
)  e.  CC )
413, 28subcld 9401 . . . . . . 7  |-  ( ph  ->  ( D  -  C
)  e.  CC )
421sigarim 27772 . . . . . . 7  |-  ( ( ( A  -  C
)  e.  CC  /\  ( D  -  C
)  e.  CC )  ->  ( ( A  -  C ) G ( D  -  C
) )  e.  RR )
4340, 41, 42syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  RR )
4443recnd 9104 . . . . 5  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  CC )
458, 28subcld 9401 . . . . . . 7  |-  ( ph  ->  ( O  -  C
)  e.  CC )
4645, 41jca 519 . . . . . 6  |-  ( ph  ->  ( ( O  -  C )  e.  CC  /\  ( D  -  C
)  e.  CC ) )
471, 46sigarimcd 27783 . . . . 5  |-  ( ph  ->  ( ( O  -  C ) G ( D  -  C ) )  e.  CC )
486, 3subcld 9401 . . . . 5  |-  ( ph  ->  ( B  -  D
)  e.  CC )
4944, 47, 48subdird 9480 . . . 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 1134 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  C  e.  CC )
)
511, 50, 15sigaradd 27787 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
521sigarperm 27781 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( C  -  O
) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C
) ) )
5328, 5, 8, 52syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
5451, 53eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( C  -  O ) G ( A  -  O ) ) )
5554oveq1d 6088 . . . 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 2469 . . 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 2472 . 2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
586, 8subcld 9401 . . . 4  |-  ( ph  ->  ( B  -  O
)  e.  CC )
591sigarac 27773 . . . 4  |-  ( ( ( B  -  O
)  e.  CC  /\  ( A  -  O
)  e.  CC )  ->  ( ( B  -  O ) G ( A  -  O
) )  =  -u ( ( A  -  O ) G ( B  -  O ) ) )
6058, 9, 59syl2anc 643 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  -u (
( A  -  O
) G ( B  -  O ) ) )
6160oveq1d 6088 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( A  -  O
) )  x.  ( C  -  D )
)  =  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D ) ) )
629, 58jca 519 . . . . 5  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
631, 62sigarimcd 27783 . . . 4  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
64 mulneg12 9462 . . . 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 643 . . 3  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  -u ( C  -  D )
) )
6628, 3negsubdi2d 9417 . . . 4  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
6766oveq2d 6089 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6865, 67eqtrd 2467 . 2  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6957, 61, 683eqtrd 2471 1  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8978   RRcr 8979   0cc0 8980    x. cmul 8985    - cmin 9281   -ucneg 9282   *ccj 11891   Imcim 11893
This theorem is referenced by:  cevath  27790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-2 10048  df-cj 11894  df-re 11895  df-im 11896
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