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Theorem cevathlem2 27858
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 968 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
4 cevath.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
54simp1d 967 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
64simp2d 968 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
73, 5, 63jca 1132 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
8 cevath.c . . . . . . . 8  |-  ( ph  ->  O  e.  CC )
95, 8subcld 9157 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  O
)  e.  CC )
103, 8subcld 9157 . . . . . . . . . 10  |-  ( ph  ->  ( D  -  O
)  e.  CC )
119, 10jca 518 . . . . . . . . 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 967 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
141, 11, 13sigariz 27853 . . . . . . . 8  |-  ( ph  ->  ( ( D  -  O ) G ( A  -  O ) )  =  0 )
158, 14jca 518 . . . . . . 7  |-  ( ph  ->  ( O  e.  CC  /\  ( ( D  -  O ) G ( A  -  O ) )  =  0 ) )
161, 7, 15sigaradd 27856 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
171sigarperm 27850 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( B  -  O
) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B
) ) )
186, 5, 8, 17syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
1916, 18eqtr4d 2318 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( B  -  O ) G ( A  -  O ) ) )
2019oveq1d 5873 . . . 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 9157 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
223, 6subcld 9157 . . . . . . 7  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2321, 22jca 518 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
241, 23sigarimcd 27852 . . . . 5  |-  ( ph  ->  ( ( A  -  B ) G ( D  -  B ) )  e.  CC )
258, 6subcld 9157 . . . . . . 7  |-  ( ph  ->  ( O  -  B
)  e.  CC )
2625, 22jca 518 . . . . . 6  |-  ( ph  ->  ( ( O  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
271, 26sigarimcd 27852 . . . . 5  |-  ( ph  ->  ( ( O  -  B ) G ( D  -  B ) )  e.  CC )
284simp3d 969 . . . . . 6  |-  ( ph  ->  C  e.  CC )
2928, 3subcld 9157 . . . . 5  |-  ( ph  ->  ( C  -  D
)  e.  CC )
3024, 27, 29subdird 9236 . . . 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 2317 . . 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 1132 . . . . 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 968 . . . . . 6  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
353, 34jca 518 . . . . 5  |-  ( ph  ->  ( D  e.  CC  /\  ( ( B  -  D ) G ( C  -  D ) )  =  0 ) )
361, 32, 35sharhght 27855 . . . 4  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( A  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
376, 28, 83jca 1132 . . . . 5  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  O  e.  CC )
)
381, 37, 35sharhght 27855 . . . 4  |-  ( ph  ->  ( ( ( O  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( O  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
3936, 38oveq12d 5876 . . 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 9157 . . . . . . 7  |-  ( ph  ->  ( A  -  C
)  e.  CC )
413, 28subcld 9157 . . . . . . 7  |-  ( ph  ->  ( D  -  C
)  e.  CC )
421sigarim 27841 . . . . . . 7  |-  ( ( ( A  -  C
)  e.  CC  /\  ( D  -  C
)  e.  CC )  ->  ( ( A  -  C ) G ( D  -  C
) )  e.  RR )
4340, 41, 42syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  RR )
4443recnd 8861 . . . . 5  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  CC )
458, 28subcld 9157 . . . . . . 7  |-  ( ph  ->  ( O  -  C
)  e.  CC )
4645, 41jca 518 . . . . . 6  |-  ( ph  ->  ( ( O  -  C )  e.  CC  /\  ( D  -  C
)  e.  CC ) )
471, 46sigarimcd 27852 . . . . 5  |-  ( ph  ->  ( ( O  -  C ) G ( D  -  C ) )  e.  CC )
486, 3subcld 9157 . . . . 5  |-  ( ph  ->  ( B  -  D
)  e.  CC )
4944, 47, 48subdird 9236 . . . 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 1132 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  C  e.  CC )
)
511, 50, 15sigaradd 27856 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
521sigarperm 27850 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( C  -  O
) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C
) ) )
5328, 5, 8, 52syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
5451, 53eqtr4d 2318 . . . . 5  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( C  -  O ) G ( A  -  O ) ) )
5554oveq1d 5873 . . . 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 2317 . . 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 2320 . 2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
586, 8subcld 9157 . . . 4  |-  ( ph  ->  ( B  -  O
)  e.  CC )
591sigarac 27842 . . . 4  |-  ( ( ( B  -  O
)  e.  CC  /\  ( A  -  O
)  e.  CC )  ->  ( ( B  -  O ) G ( A  -  O
) )  =  -u ( ( A  -  O ) G ( B  -  O ) ) )
6058, 9, 59syl2anc 642 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  -u (
( A  -  O
) G ( B  -  O ) ) )
6160oveq1d 5873 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( A  -  O
) )  x.  ( C  -  D )
)  =  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D ) ) )
629, 58jca 518 . . . . 5  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
631, 62sigarimcd 27852 . . . 4  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
64 mulneg12 9218 . . . 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 642 . . 3  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  -u ( C  -  D )
) )
6628, 3negsubdi2d 9173 . . . 4  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
6766oveq2d 5874 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6865, 67eqtrd 2315 . 2  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6957, 61, 683eqtrd 2319 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    - cmin 9037   -ucneg 9038   *ccj 11581   Imcim 11583
This theorem is referenced by:  cevath  27859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586
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