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Theorem sigarac 29888
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 29886 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
3 cjcl 12594 . . . . . . 7  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
43adantl 466 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  B
)  e.  CC )
5 simpl 457 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
64, 5cjmuld 12710 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
( * `  B
)  x.  A ) )  =  ( ( * `  ( * `
 B ) )  x.  ( * `  A ) ) )
7 simpr 461 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
87cjcjd 12688 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
98oveq1d 6106 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  ( * `  B
) )  x.  (
* `  A )
)  =  ( B  x.  ( * `  A ) ) )
105cjcld 12685 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  A
)  e.  CC )
117, 10mulcomd 9407 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  x.  (
* `  A )
)  =  ( ( * `  A )  x.  B ) )
126, 9, 113eqtrrd 2480 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  B
)  =  ( * `
 ( ( * `
 B )  x.  A ) ) )
1312fveq2d 5695 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  (
( * `  A
)  x.  B ) )  =  ( Im
`  ( * `  ( ( * `  B )  x.  A
) ) ) )
144, 5mulcld 9406 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  B )  x.  A
)  e.  CC )
1514imcjd 12694 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  (
* `  ( (
* `  B )  x.  A ) ) )  =  -u ( Im `  ( ( * `  B )  x.  A
) ) )
162, 13, 153eqtrd 2479 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u (
Im `  ( (
* `  B )  x.  A ) ) )
171sigarval 29886 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B G A )  =  ( Im
`  ( ( * `
 B )  x.  A ) ) )
1817ancoms 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B G A )  =  ( Im
`  ( ( * `
 B )  x.  A ) ) )
1918negeqd 9604 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B G A )  =  -u ( Im `  ( ( * `  B )  x.  A ) ) )
2016, 19eqtr4d 2478 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  -u ( B G A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280    x. cmul 9287   -ucneg 9596   *ccj 12585   Imcim 12587
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 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-2 10380  df-cj 12588  df-re 12589  df-im 12590
This theorem is referenced by:  sigaras  29891  sigarms  29892  sigarperm  29896  sigariz  29899  sigarcol  29900  sigaradd  29902  cevathlem2  29904
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