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Theorem sigarac 27818
Description: Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
Hypothesis
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
Assertion
Ref Expression
sigarac  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarac
StepHypRef Expression
1 sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
21sigarval 27816 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
3 cjcl 11910 . . . . . . 7  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
43adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
5 simpl 444 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
64, 5cjmuld 12026 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
( * `  B
)  x.  A ) )  =  ( ( * `  ( * `
 B ) )  x.  ( * `  A ) ) )
7 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
87cjcjd 12004 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
98oveq1d 6096 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  ( * `  B
) )  x.  (
* `  A )
)  =  ( B  x.  ( * `  A ) ) )
105cjcld 12001 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
117, 10mulcomd 9109 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  (
* `  A )
)  =  ( ( * `  A )  x.  B ) )
126, 9, 113eqtrrd 2473 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  B
)  =  ( * `
 ( ( * `
 B )  x.  A ) ) )
1312fveq2d 5732 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  (
( * `  A
)  x.  B ) )  =  ( Im
`  ( * `  ( ( * `  B )  x.  A
) ) ) )
144, 5mulcld 9108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  B )  x.  A
)  e.  CC )
1514imcjd 12010 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  (
* `  ( (
* `  B )  x.  A ) ) )  =  -u ( Im `  ( ( * `  B )  x.  A
) ) )
162, 13, 153eqtrd 2472 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u (
Im `  ( (
* `  B )  x.  A ) ) )
171sigarval 27816 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  =  ( Im
`  ( ( * `
 B )  x.  A ) ) )
1817ancoms 440 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B G A )  =  ( Im
`  ( ( * `
 B )  x.  A ) ) )
1918negeqd 9300 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B G A )  =  -u ( Im `  ( ( * `  B )  x.  A ) ) )
2016, 19eqtr4d 2471 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988    x. cmul 8995   -ucneg 9292   *ccj 11901   Imcim 11903
This theorem is referenced by:  sigaras  27821  sigarms  27822  sigarperm  27826  sigariz  27829  sigarcol  27830  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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