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Theorem cevathlem2 29909
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 1001 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
4 cevath.a . . . . . . . . 9  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
54simp1d 1000 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
64simp2d 1001 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
73, 5, 63jca 1168 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
8 cevath.c . . . . . . . 8  |-  ( ph  ->  O  e.  CC )
95, 8subcld 9724 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  O
)  e.  CC )
103, 8subcld 9724 . . . . . . . . . 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 1000 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
141, 11, 13sigariz 29904 . . . . . . . 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 29907 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
171sigarperm 29901 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( B  -  O
) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B
) ) )
186, 5, 8, 17syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( ( B  -  O ) G ( A  -  O ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
1916, 18eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( B  -  O ) G ( A  -  O ) ) )
2019oveq1d 6111 . . . 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 9724 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
223, 6subcld 9724 . . . . . . 7  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2321, 22jca 532 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
241, 23sigarimcd 29903 . . . . 5  |-  ( ph  ->  ( ( A  -  B ) G ( D  -  B ) )  e.  CC )
258, 6subcld 9724 . . . . . . 7  |-  ( ph  ->  ( O  -  B
)  e.  CC )
2625, 22jca 532 . . . . . 6  |-  ( ph  ->  ( ( O  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
271, 26sigarimcd 29903 . . . . 5  |-  ( ph  ->  ( ( O  -  B ) G ( D  -  B ) )  e.  CC )
284simp3d 1002 . . . . . 6  |-  ( ph  ->  C  e.  CC )
2928, 3subcld 9724 . . . . 5  |-  ( ph  ->  ( C  -  D
)  e.  CC )
3024, 27, 29subdird 9806 . . . 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 2477 . . 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 1168 . . . . 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 1001 . . . . . 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 29906 . . . 4  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( A  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
376, 28, 83jca 1168 . . . . 5  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  O  e.  CC )
)
381, 37, 35sharhght 29906 . . . 4  |-  ( ph  ->  ( ( ( O  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( O  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
3936, 38oveq12d 6114 . . 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 9724 . . . . . . 7  |-  ( ph  ->  ( A  -  C
)  e.  CC )
413, 28subcld 9724 . . . . . . 7  |-  ( ph  ->  ( D  -  C
)  e.  CC )
421sigarim 29892 . . . . . . 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 9417 . . . . 5  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  CC )
458, 28subcld 9724 . . . . . . 7  |-  ( ph  ->  ( O  -  C
)  e.  CC )
4645, 41jca 532 . . . . . 6  |-  ( ph  ->  ( ( O  -  C )  e.  CC  /\  ( D  -  C
)  e.  CC ) )
471, 46sigarimcd 29903 . . . . 5  |-  ( ph  ->  ( ( O  -  C ) G ( D  -  C ) )  e.  CC )
486, 3subcld 9724 . . . . 5  |-  ( ph  ->  ( B  -  D
)  e.  CC )
4944, 47, 48subdird 9806 . . . 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 1168 . . . . . . 7  |-  ( ph  ->  ( D  e.  CC  /\  A  e.  CC  /\  C  e.  CC )
)
511, 50, 15sigaradd 29907 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
521sigarperm 29901 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  O  e.  CC )  ->  (
( C  -  O
) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C
) ) )
5328, 5, 8, 52syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
5451, 53eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( C  -  O ) G ( A  -  O ) ) )
5554oveq1d 6111 . . . 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 2477 . . 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 2480 . 2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
586, 8subcld 9724 . . . 4  |-  ( ph  ->  ( B  -  O
)  e.  CC )
591sigarac 29893 . . . 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 6111 . 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 29903 . . . 4  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
64 mulneg12 9788 . . . 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 9740 . . . 4  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
6766oveq2d 6112 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6865, 67eqtrd 2475 . 2  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6957, 61, 683eqtrd 2479 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   CCcc 9285   RRcr 9286   0cc0 9287    x. cmul 9292    - cmin 9600   -ucneg 9601   *ccj 12590   Imcim 12592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-cj 12593  df-re 12594  df-im 12595
This theorem is referenced by:  cevath  29910
  Copyright terms: Public domain W3C validator