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Theorem sigarperm 32316
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 995 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 996 . . . . . 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 32307 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  RR )
54recnd 9611 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
61, 2, 5syl2anc 659 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
7 simp1 994 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
83sigarim 32307 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
98recnd 9611 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  CC )
101, 7, 9syl2anc 659 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
116, 10negsubd 9928 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  -  ( B G A ) ) )
123sigarac 32308 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
137, 1, 12syl2anc 659 . . . . . 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 6286 . . . 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 6285 . 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 32315 . . 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 1202 . 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 32315 . . 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 32307 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
227, 1, 21syl2anc 659 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  RR )
2322recnd 9611 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  CC )
243sigarim 32307 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
257, 2, 24syl2anc 659 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
2625recnd 9611 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  CC )
273sigarim 32307 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
282, 1, 27syl2anc 659 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  RR )
2928recnd 9611 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
3023, 26, 29sub32d 9954 . . 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 9771 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  ( A G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
323sigarac 32308 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  =  -u ( C G B ) )
331, 2, 32syl2anc 659 . . . . . . 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 6286 . . . . 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 9928 . . . . 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 6285 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   CCcc 9479   RRcr 9480    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797   *ccj 13011   Imcim 13013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-cj 13014  df-re 13015  df-im 13016
This theorem is referenced by:  sigarcol  32320  sharhght  32321  sigaradd  32322  cevathlem2  32324
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