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Theorem sigarperm 27953
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 956 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
2 simp3 957 . . . . . 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 27944 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  RR )
54recnd 8877 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
61, 2, 5syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  e.  CC )
7 simp1 955 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
83sigarim 27944 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  RR )
98recnd 8877 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  e.  CC )
101, 7, 9syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G A )  e.  CC )
116, 10negsubd 9179 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  -u ( B G A ) )  =  ( ( B G C )  -  ( B G A ) ) )
123sigarac 27945 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
137, 1, 12syl2anc 642 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  = 
-u ( B G A ) )
1413eqcomd 2301 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( B G A )  =  ( A G B ) )
1514oveq2d 5890 . . . 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 2330 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  -  ( B G A ) )  =  ( ( B G C )  +  ( A G B ) ) )
1716oveq1d 5889 . 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 27952 . . 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 1159 . 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 27952 . . 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 27944 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  e.  RR )
227, 1, 21syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  RR )
2322recnd 8877 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G B )  e.  CC )
243sigarim 27944 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
257, 2, 24syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  RR )
2625recnd 8877 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A G C )  e.  CC )
273sigarim 27944 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C G B )  e.  RR )
282, 1, 27syl2anc 642 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  RR )
2928recnd 8877 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C G B )  e.  CC )
3023, 26, 29sub32d 9205 . . 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 9030 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B G C )  +  ( A G B ) )  =  ( ( A G B )  +  ( B G C ) ) )
323sigarac 27945 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  =  -u ( C G B ) )
331, 2, 32syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B G C )  = 
-u ( C G B ) )
3433eqcomd 2301 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  -u ( C G B )  =  ( B G C ) )
3534oveq2d 5890 . . . . 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 9179 . . . . 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 2335 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A G B )  -  ( C G B ) )  =  ( ( B G C )  +  ( A G B ) ) )
3837oveq1d 5889 . . 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 2332 . 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 2339 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054   *ccj 11597   Imcim 11599
This theorem is referenced by:  sigarcol  27957  sharhght  27958  sigaradd  27959  cevathlem2  27961
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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