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Theorem cevathlem2 27961
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 9173 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  O
)  e.  CC )
103, 8subcld 9173 . . . . . . . . . 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 27956 . . . . . . . 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 27959 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( A  -  B ) G ( O  -  B ) ) )
171sigarperm 27953 . . . . . . 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 2331 . . . . 5  |-  ( ph  ->  ( ( ( A  -  B ) G ( D  -  B
) )  -  (
( O  -  B
) G ( D  -  B ) ) )  =  ( ( B  -  O ) G ( A  -  O ) ) )
2019oveq1d 5889 . . . 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 9173 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
223, 6subcld 9173 . . . . . . 7  |-  ( ph  ->  ( D  -  B
)  e.  CC )
2321, 22jca 518 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
241, 23sigarimcd 27955 . . . . 5  |-  ( ph  ->  ( ( A  -  B ) G ( D  -  B ) )  e.  CC )
258, 6subcld 9173 . . . . . . 7  |-  ( ph  ->  ( O  -  B
)  e.  CC )
2625, 22jca 518 . . . . . 6  |-  ( ph  ->  ( ( O  -  B )  e.  CC  /\  ( D  -  B
)  e.  CC ) )
271, 26sigarimcd 27955 . . . . 5  |-  ( ph  ->  ( ( O  -  B ) G ( D  -  B ) )  e.  CC )
284simp3d 969 . . . . . 6  |-  ( ph  ->  C  e.  CC )
2928, 3subcld 9173 . . . . 5  |-  ( ph  ->  ( C  -  D
)  e.  CC )
3024, 27, 29subdird 9252 . . . 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 2330 . . 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 27958 . . . 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 27958 . . . 4  |-  ( ph  ->  ( ( ( O  -  B ) G ( D  -  B
) )  x.  ( C  -  D )
)  =  ( ( ( O  -  C
) G ( D  -  C ) )  x.  ( B  -  D ) ) )
3936, 38oveq12d 5892 . . 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 9173 . . . . . . 7  |-  ( ph  ->  ( A  -  C
)  e.  CC )
413, 28subcld 9173 . . . . . . 7  |-  ( ph  ->  ( D  -  C
)  e.  CC )
421sigarim 27944 . . . . . . 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 8877 . . . . 5  |-  ( ph  ->  ( ( A  -  C ) G ( D  -  C ) )  e.  CC )
458, 28subcld 9173 . . . . . . 7  |-  ( ph  ->  ( O  -  C
)  e.  CC )
4645, 41jca 518 . . . . . 6  |-  ( ph  ->  ( ( O  -  C )  e.  CC  /\  ( D  -  C
)  e.  CC ) )
471, 46sigarimcd 27955 . . . . 5  |-  ( ph  ->  ( ( O  -  C ) G ( D  -  C ) )  e.  CC )
486, 3subcld 9173 . . . . 5  |-  ( ph  ->  ( B  -  D
)  e.  CC )
4944, 47, 48subdird 9252 . . . 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 27959 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( A  -  C ) G ( O  -  C ) ) )
521sigarperm 27953 . . . . . . 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 2331 . . . . 5  |-  ( ph  ->  ( ( ( A  -  C ) G ( D  -  C
) )  -  (
( O  -  C
) G ( D  -  C ) ) )  =  ( ( C  -  O ) G ( A  -  O ) ) )
5554oveq1d 5889 . . . 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 2330 . . 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 2333 . 2  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( B  -  O
) G ( A  -  O ) )  x.  ( C  -  D ) ) )
586, 8subcld 9173 . . . 4  |-  ( ph  ->  ( B  -  O
)  e.  CC )
591sigarac 27945 . . . 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 5889 . 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 27955 . . . 4  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
64 mulneg12 9234 . . . 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 9189 . . . 4  |-  ( ph  -> 
-u ( C  -  D )  =  ( D  -  C ) )
6766oveq2d 5890 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  -u ( C  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6865, 67eqtrd 2328 . 2  |-  ( ph  ->  ( -u ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) )
6957, 61, 683eqtrd 2332 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 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    - cmin 9053   -ucneg 9054   *ccj 11597   Imcim 11599
This theorem is referenced by:  cevath  27962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602
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