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Theorem sigarperm 27826
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 958 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 959 . . . . . 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 27817 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  RR )
54recnd 9114 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
61, 2, 5syl2anc 643 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
7 simp1 957 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
83sigarim 27817 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
98recnd 9114 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  CC )
101, 7, 9syl2anc 643 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
116, 10negsubd 9417 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  -  ( B G A ) ) )
123sigarac 27818 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
137, 1, 12syl2anc 643 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
1413eqcomd 2441 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
1514oveq2d 6097 . . . 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 2470 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  -  ( B G A ) )  =  ( ( B G C )  +  ( A G B ) ) )
1716oveq1d 6096 . 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 27825 . . 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 1161 . 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 27825 . . 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 27817 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
227, 1, 21syl2anc 643 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  RR )
2322recnd 9114 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  CC )
243sigarim 27817 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
257, 2, 24syl2anc 643 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
2625recnd 9114 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  CC )
273sigarim 27817 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
282, 1, 27syl2anc 643 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  RR )
2928recnd 9114 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
3023, 26, 29sub32d 9443 . . 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 9268 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  ( A G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
323sigarac 27818 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  =  -u ( C G B ) )
331, 2, 32syl2anc 643 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  = 
-u ( C G B ) )
3433eqcomd 2441 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G B )  =  ( B G C ) )
3534oveq2d 6097 . . . . 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 9417 . . . . 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 2475 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  -  ( C G B ) )  =  ( ( B G C )  +  ( A G B ) ) )
3837oveq1d 6096 . . 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 2472 . 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 2479 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
) G ( B  -  C ) )  =  ( ( B  -  A ) G ( C  -  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   RRcr 8989    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   *ccj 11901   Imcim 11903
This theorem is referenced by:  sigarcol  27830  sharhght  27831  sigaradd  27832  cevathlem2  27834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-cj 11904  df-re 11905  df-im 11906
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