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Theorem sigarperm 30043
Description: Signed area  ( A  -  C ) G ( B  -  C
) acts as a double area of a triangle  A B C. Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)
Hypothesis
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
Assertion
Ref Expression
sigarperm  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A
) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarperm
StepHypRef Expression
1 simp2 989 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 990 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 sigar . . . . . . . 8  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
43sigarim 30034 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  RR )
54recnd 9522 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
61, 2, 5syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
7 simp1 988 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
83sigarim 30034 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
98recnd 9522 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  CC )
101, 7, 9syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
116, 10negsubd 9835 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  -  ( B G A ) ) )
123sigarac 30035 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
137, 1, 12syl2anc 661 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
1413eqcomd 2462 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
1514oveq2d 6215 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  +  ( A G B ) ) )
1611, 15eqtr3d 2497 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  -  ( B G A ) )  =  ( ( B G C )  +  ( A G B ) ) )
1716oveq1d 6214 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( B G C )  -  ( B G A ) )  -  ( A G C ) )  =  ( ( ( B G C )  +  ( A G B ) )  -  ( A G C ) ) )
183sigarexp 30042 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( B  -  A
) G ( C  -  A ) )  =  ( ( ( B G C )  -  ( B G A ) )  -  ( A G C ) ) )
19183comr 1196 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  -  A
) G ( C  -  A ) )  =  ( ( ( B G C )  -  ( B G A ) )  -  ( A G C ) ) )
203sigarexp 30042 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( ( A G B )  -  ( A G C ) )  -  ( C G B ) ) )
213sigarim 30034 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
227, 1, 21syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  RR )
2322recnd 9522 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  CC )
243sigarim 30034 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
257, 2, 24syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
2625recnd 9522 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  CC )
273sigarim 30034 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
282, 1, 27syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  RR )
2928recnd 9522 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
3023, 26, 29sub32d 9861 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A G B )  -  ( A G C ) )  -  ( C G B ) )  =  ( ( ( A G B )  -  ( C G B ) )  -  ( A G C ) ) )
316, 23addcomd 9681 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  ( A G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
323sigarac 30035 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  =  -u ( C G B ) )
331, 2, 32syl2anc 661 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  = 
-u ( C G B ) )
3433eqcomd 2462 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G B )  =  ( B G C ) )
3534oveq2d 6215 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  +  -u ( C G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
3623, 29negsubd 9835 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  +  -u ( C G B ) )  =  ( ( A G B )  -  ( C G B ) ) )
3731, 35, 363eqtr2rd 2502 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  -  ( C G B ) )  =  ( ( B G C )  +  ( A G B ) ) )
3837oveq1d 6214 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A G B )  -  ( C G B ) )  -  ( A G C ) )  =  ( ( ( B G C )  +  ( A G B ) )  -  ( A G C ) ) )
3920, 30, 383eqtrd 2499 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( ( B G C )  +  ( A G B ) )  -  ( A G C ) ) )
4017, 19, 393eqtr4rd 2506 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   CCcc 9390   RRcr 9391    + caddc 9395    x. cmul 9397    - cmin 9705   -ucneg 9706   *ccj 12702   Imcim 12704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-2 10490  df-cj 12705  df-re 12706  df-im 12707
This theorem is referenced by:  sigarcol  30047  sharhght  30048  sigaradd  30049  cevathlem2  30051
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