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Theorem sigarperm 29867
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 29858 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  RR )
54recnd 9404 . . . . . 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 29858 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
98recnd 9404 . . . . . 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 9717 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  -  ( B G A ) ) )
123sigarac 29859 . . . . . . 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 2443 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
1514oveq2d 6102 . . . 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 2472 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  -  ( B G A ) )  =  ( ( B G C )  +  ( A G B ) ) )
1716oveq1d 6101 . 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 29866 . . 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 1195 . 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 29866 . . 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 29858 . . . . . 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 9404 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  CC )
243sigarim 29858 . . . . . 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 9404 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  CC )
273sigarim 29858 . . . . . 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 9404 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
3023, 26, 29sub32d 9743 . . 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 9563 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  ( A G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
323sigarac 29859 . . . . . . . 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 2443 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G B )  =  ( B G C ) )
3534oveq2d 6102 . . . . 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 9717 . . . . 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 2477 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  -  ( C G B ) )  =  ( ( B G C )  +  ( A G B ) ) )
3837oveq1d 6101 . . 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 2474 . 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 2481 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 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   CCcc 9272   RRcr 9273    + caddc 9277    x. cmul 9279    - cmin 9587   -ucneg 9588   *ccj 12577   Imcim 12579
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-cj 12580  df-re 12581  df-im 12582
This theorem is referenced by:  sigarcol  29871  sharhght  29872  sigaradd  29873  cevathlem2  29875
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